Differential Geometry: Bundles, Connections, Metrics and Curvature

Oxford Graduate Texts in Mathematics Series Editors R. Cohen S. K. Donaldson S. Hildebrandt T. J. Lyons M. J. Taylor

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Keith Hannabuss: An Introduction to Quantum Theory Reinhold Meise and Dietmar Vogt: Introduction to Functional Analysis James G. Oxley: Matroid Theory N.J. Hitchin, G.B. Segal, and R.S. Ward: Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces Wulf Rossmann: Lie Groups: An Introduction through Linear Groups Qing Liu: Algebraic Geometry and Arithmetic Curves Martin R. Bridson and Simon M. Salamon (eds): Invitations to Geometry and Topology Shmuel Kantorovitz: Introduction to Modern Analysis Terry Lawson: Topology: A Geometric Approach Meinolf Geek: An Introduction to Algebraic Geometry and Algebraic Groups Alastair Fletcher and Vladimir Markovic: Quasiconformal Maps and Teichmu¨ller Theory Dominic Joyce: Riemannian Holonomy Groups and Calibrated Geometry Fernando Villegas: Experimental Number Theory Pe´ter Medvegyev: Stochastic Integration Theory Martin A. Guest: From Quantum Cohomology to Integrable Systems Alan D. Rendall: Partial Differential Equations in General Relativity Yves Fe´lix, John Oprea and Daniel Tanre´: Algebraic Models in Geometry Jie Xiong: Introduction to Stochastic Filtering Theory Maciej Dunajski: Solitons, Instantons, and Twistors Graham R. Allan: Introduction to Banach Spaces and Algebras James Oxley: Matroid Theory, Second Edition Simon Donaldson: Riemann Surfaces Clifford Henry Taubes: Differential Geometry: Bundles, Connections, Metrics and Curvature

Differential Geometry Bundles, Connections, Metrics and Curvature

Clifford Henry Taubes Department of Mathematics Harvard University

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Great Clarendon Street, Oxford ox2 6dp Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York # Clifford Henry Taubes 2011 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2011 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by SPI Publisher Services, Pondicherry, India Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY ISBN 978–0–19–960588–0 (hbk.) 978–0–19–960587–3 (pbk.) 1 3 5 7 9 10 8 6 4 2

Preface

This book is meant to be an introduction to the subject of vector bundles, principal bundles, metrics (Riemannian and otherwise), covariant derivatives, connections and curvature. I am imagining an audience of first-year graduate students or advanced undergraduate students who have some familiarity with the basics of linear algebra and with the notion of a smooth manifold. Even so, I start with a review of the latter subject. I have tried to make the presentation as much as possible self-contained with proofs of basic results presented in full. In particular, I have supplied proofs for almost all of the background material either in the text or in the chapter appendices. Even so, you will most likely have trouble if you are not accustomed to matrices with real and complex number entries, in particular the notions of an eigenvalue and eigenvector. You should also be comfortable working with multi-variable calculus. At the very end of each chapter is a very brief list of other books with parts that cover some of the chapter’s subject matter. I have worked out many examples in the text, because truth be told, the subject is not interesting to me in the abstract. I for one need to feel the geometry to understand what is going on. In particular, I present in detail many of the foundational examples. I learned much of the material that I present here from a true master, Raoul Bott. In particular, I put into this book the topics that I recall Raoul covering in his first-semester graduate differential geometry class. Although the choice of topics are those I recall Raoul covering, the often idionsyncratic points of view and the exposition are my own.

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Contents

1

2

3

Smooth manifolds

1

1.1 1.2 1.3 1.4 1.5 1.6

1 3 4 7 8 9

Smooth manifolds The inverse function theorem and implicit function theorem Submanifolds of Rm Submanifolds of manifolds More constructions of manifolds More smooth manifolds: The Grassmannians Appendix 1.1 How to prove the inverse function and implicit function theorems Appendix 1.2 Partitions of unity Additional reading

11 13 13

Matrices and Lie groups

14

2.1 2.2 2.3 2.4 2.5 2.6

14 15 16 17 19 21 22 24

The general linear group Lie groups Examples of Lie groups Some complex Lie groups The groups Sl(n; C), U(n) and SU(n) Notation with regards to matrices and differentials Appendix 2.1 The transition functions for the Grassmannians Additional reading

Introduction to vector bundles

25

3.1 3.2 3.3 3.4

25 27 28 29

The definition The standard definition The first examples of vector bundles The tangent bundle

viii

Contents

3.5 3.6 3.7 3.8 3.9

Tangent bundle examples The cotangent bundle Bundle homomorphisms Sections of vector bundles Sections of TM and T*M Additional reading

4 Algebra of vector bundles 4.1 4.2 4.3 4.4 4.5 4.6 4.7

Subbundles Quotient bundles The dual bundle Bundles of homomorphisms Tensor product bundles The direct sum Tensor powers Additional reading

5 Maps and vector bundles 5.1 The pull-back construction 5.2 Pull-backs and Grassmannians 5.3 Pull-back of differential forms and push-forward of vector fields 5.4 Invariant forms and vector fields on Lie groups 5.5 The exponential map on a matrix group 5.6 The exponential map and right/left invariance on Gl(n; C) and its subgroups 5.7 Immersion, submersion and transversality Additional reading 6 Vector bundles with Cn as fiber 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8

Definitions Comparing definitions Examples: The complexification Complex bundles over surfaces in R3 The tangent bundle to a surface in R3 Bundles over 4-dimensional submanifolds in R5 Complex bundles over 4-dimensional manifolds Complex Grassmannians

31 33 34 35 36 38 39 39 40 41 42 43 43 44 46 48 48 49 50 52 53 55 57 58 59 59 60 62 63 64 64 65 65

Contents

6.9 The exterior product construction 6.10 Algebraic operations 6.11 Pull-back Additional reading 7 Metrics on vector bundles 7.1 Metrics and transition functions for real vector bundles 7.2 Metrics and transition functions for complex vector bundles 7.3 Metrics, algebra and maps 7.4 Metrics on TM Additional reading 8 Geodesics 8.1 8.2 8.3 8.4 8.5 8.6 8.7

Riemannian metrics and distance Length minimizing curves The existence of geodesics First examples Geodesics on SO(n) Geodesics on U(n) and SU(n) Geodesics and matrix groups Appendix 8.1 The proof of the vector field theorem Additional reading

9 Properties of geodesics 9.1 9.2 9.3 9.4

10

The maximal extension of a geodesic The exponential map Gaussian coordinates The proof of the geodesic theorem Additional reading

68 69 70 71 72 73 75 75 77 77 78 78 79 81 82 85 89 92 93 94 96 96 96 98 100 103

Principal bundles

104

10.1 10.2 10.3 10.4

104 105 106 108

The definition A cocycle definition Principal bundles constructed from vector bundles Quotients of Lie groups by subgroups

ix

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Contents

10.5 10.6 10.7 10.8 10.9

Examples of Lie group quotients Cocycle construction examples Pull-backs of principal bundles Reducible principal bundles Associated vector bundles Appendix 10.1 Proof of Proposition 10.1 Additional reading

11 Covariant derivatives and connections 11.1 11.2 11.3 11.4 11.5 11.6 11.7

Covariant derivatives The space of covariant derivatives Another construction of covariant derivatives Principal bundles and connections Connections and covariant derivatives Horizontal lifts An application to the classification of principal G-bundles up to isomorphism 11.8 Connections, covariant derivatives and pull-back bundles Additional reading

12 Covariant derivatives, connections and curvature 12.1 Exterior derivative 12.2 Closed forms, exact forms, diffeomorphisms and De Rham cohomology 12.3 Lie derivative 12.4 Curvature and covariant derivatives 12.5 An example 12.6 Curvature and commutators 12.7 Connections and curvature 12.8 The horizontal subbundle revisited Additional reading 13 Flat connections and holonomy 13.1 13.2 13.3 13.4 13.5

Flat connections Flat connections on bundles over the circle Foliations Automorphisms of a principal bundle The fundamental group

110 113 116 118 119 121 124 125 125 126 127 128 134 135 136 137 138 139 139 141 143 144 146 148 148 150 151 152 152 153 155 156 157

Contents

13.6 13.7 13.8 13.9

14

170

14.1 14.2 14.3 14.4 14.5

170 171 174 175

14.7 14.8 14.9

16

159 159 160 162 164 166 169

Curvature polynomials and characteristic classes

14.6

15

The flat connections on bundles over M The universal covering space Holonomy and curvature Proof of the classification theorem for flat connections Appendix 13.1 Smoothing maps Appendix 13.2 The proof of the Frobenius theorem Additional reading

The Bianchi Identity Characteristic forms Characteristic classes: Part 1 Characteristic classes: Part 2 Characteristic classes for complex vector bundles and the Chern classes Characteristic classes for real vector bundles and the Pontryagin classes Examples of bundles with nonzero Chern classes The degree of the map g ! gm from SU(2) to itself A Chern–Simons form Appendix 14.1 The ad-invariant functions on M(n; C) Appendix 14.2 Integration on manifolds Appendix 14.3 The degree of a map Additional reading

177 179 180 189 190 190 192 197 204

Covariant derivatives and metrics

205

15.1 15.2 15.3 15.4 15.5 15.6 15.7

205 208 210 211 212 214 216 218

Metric compatible covariant derivatives Torsion free covariant derivatives on T*M The Levi-Civita connection/covariant derivative A formula for the Levi-Civita connection Covariantly constant sections An example of the Levi-Civita connection The curvature of the Levi-Civita connection Additional reading

The Riemann curvature tensor

220

16.1 Spherical metrics, flat metrics and hyperbolic metrics 16.2 The Schwarzchild metric

220 223

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Contents

16.3 16.4 16.5 16.6 16.7 16.8 16.9

Curvature conditions Manifolds of dimension 2: The Gauss–Bonnet formula Metrics on manifolds of dimension 2 Conformal changes Sectional curvatures and universal covering spaces The Jacobi field equation Constant sectional curvature and the Jacobi field equation 16.10 Manifolds of dimension 3 16.11 The Riemannian curvature of a compact matrix group Additional reading

17 Complex manifolds 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 17.9 17.10

Some basics concerning holomorphic functions on Cn The definition of a complex manifold First examples of complex manifolds The Newlander–Nirenberg theorem Metrics and almost complex structures on TM The almost Ka¨hler 2-form Symplectic forms Ka¨hler manifolds Complex manifolds with closed almost Ka¨hler form Examples of Ka¨hler manifolds Appendix 17.1 Compatible almost complex structures Additional reading

18 Holomorphic submanifolds, holomorphic sections and curvature 18.1 18.2 18.3 18.4 18.5 18.6 18.7

Holomorphic submanifolds of a complex manifold Holomorphic submanifolds of projective spaces Proof of Proposition 18.2, about holomorphic submanifolds in CPn The curvature of a Ka¨hler metric Curvature with no (0, 2) part Holomorphic sections Example on CPn Additional reading

224 227 229 230 232 233 236 238 239 244 245 246 247 248 251 255 255 256 257 258 259 261 267

268 268 269 271 272 275 277 279 281

Contents

19

The Hodge star

282

19.1 19.2 19.3 19.4 19.5

282 283 284 285 286 287

Definition of the Hodge star Representatives of De Rham cohomology A fairy tale The Hodge theorem Self-duality Additional reading

List of lemmas, propositions, corollaries and theorems List of symbols Index

289 291 295

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1

Smooth manifolds

Said briefly, differential geometry studies various constructions that can be built on a smooth manifold. Granted this, the task for Chapter 1 is to summarize various facts about smooth manifolds that are used, either explicitly or implicitly, in much of what follows. These facts are offered sans proof since proofs can be had in most any textbook on differential topology; my favorite being Differential Topology by Victor Guillemin and Alan Pollack (1974).

1.1

Smooth manifolds

The Ur smooth manifold of dimension n is the n-dimensional Euclidean space, Rn. I think of Rn as coming with a standard set of coordinates, (x1, . . . , xn). The Euclidean space is the local model for all other manifolds.

1.1.1

Topological manifolds

A manifold of dimension n is a paracompact, Hausdorff space such that each point has a neighborhood that is homeomorphic to Rn or, what amounts to the same thing, to the interior of a ball in Rn. Such a neighborhood is called a coordinate chart and the pull-back of the coordinate functions from Rn are called local coordinates. A collection U ¼ fðU; jÞ : U  M is open and j: U ! Rn is an embeddingg



is said to be a coordinate atlas for M if M ¼ (U,j)2U U. A coordinate atlas is said to be locally finite if it is the case that each point p 2 M is contained in a finite collection of its open sets. Two topological manifolds, M and M0 , are said to be homeomorphic if there is a homeomorphism between them. This is a continuous, 1–1, surjective map from M to M0 whose inverse is continuous.

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1.1.2

Smooth manifolds

Let U denote a coordinate atlas for M. Suppose that (U, jU) and (U0 , jU0 ) are two elements from U. The map ¨

0 jU0  j1 U : jU ðU

UÞ ! jU0 ðU 0

¨

UÞ

is a homeomorphism between two open subsets of Euclidean space. This map is said to be the coordinate transition function for the pair of charts (U, jU) and (U0 , jU0 ). A smooth structure on M is defined by an equivalence class of coordinate atlases with the following property: All transition functions are diffeomorphisms. This is to say that they have partial derivatives to all orders, as do their inverses. Coordinate atlases U and V are deemed to be equivalent when the following condition holds: Given any pairs (U, jU) 2 U and (V, jV) 2 V, the compositions jU  jV1: jV ðV UÞ ! jU ðV UÞ; jV  jU1: jU ðV UÞ ! jV ðV UÞ ¨

¨

¨

¨

ð*Þ

are diffeomorphisms between open subsets of Euclidean space. In what follows, a map with partial derivatives to all orders is said to be smooth. A manifold M with a smooth structure is said to be a smooth manifold. The point here is that one can do calculus on smooth manifolds, for it makes sense to say that a function f: M ! R is differentiable, or infinitely differentiable. The collection of smooth functions on M is denoted by C1(M; R) and consists of those functions with the following property: Let U denote any given coordinate atlas from the equivalence class that defines the smooth structure. If (U, j) 2 U, then f  j1 is a smooth function on j(U). This requirement defines the same set of smooth functions no matter the choice of representative atlas by virtue of the condition (*) that defines the equivalence relation.

1.1.3

Maps between smooth manifolds

If M and N are smooth manifolds, then a map h: M ! N is said to be smooth if the following is true: Let U denote a locally finite coordinate atlas from the equivalence class that gives the smooth structure to M; and let V denote a corresponding atlas for N. Then each map in the collection fc  h  j1 : hðUÞ

¨

2

V 6¼ gðU;jÞ2U;ðV;cÞ2V

is smooth as a map from one Euclidean space to another. Note again that the equivalence relation described by (*) above guarantees that the definition of a smooth map depends only on the smooth structures of M and N, but not on the chosen representative coordinate atlases.

1.2

The inverse function theorem and implicit function theorem

Two smooth manifolds are said to be diffeomorphic when there exists a smooth homeomorphism h: M ! N with smooth inverse. This is to say that M is homeomorphic to N and that the given equivalence classes of atlases that define the respective smooth structures are one and the same.

1.2

The inverse function theorem and implicit function theorem

Various constructions of smooth manifolds rely on two theorems from differential topology, these being the inverse function theorem and the implicit function theorem. What follows here is a digression to state these theorems. Their proofs are sketched in Appendix 1.1. The statement of these theorems introduces the notion of the differential of a map between one Euclidean space and another. Here is a reminder: Let U  Rm denote a given open set and let c: U ! Rn denote a given map. The differential of c is denoted by c*; it is a matrix valued function on U with m columns and n rows whose entries come from the partial derivatives of the components of c. This is to say that if c has components (c1, . . . , cn), then the entry in the i’th i row and j’th column is @j ci ¼ @c @xj : (Here and throughout, @ j is shorthand for the partial derivative with respect to the coordinate xj.) Theorem 1.1 (the inverse function theorem): Let U  Rm denote a neighborhood of the origin, and let c: U ! Rm denote a smooth map. Suppose that p 2 U and that the matrix c* at p is invertible. Then there is a neighborhood V  Rm of c(p) and a smooth map s: V ! U such that s(c(p)) ¼ p and  s  c is the identity on some neighborhood U0  U of p,  c  s is the identity on V. Conversely, if c* is not invertible at p, then there is no such set V and map s with these properties. The key conclusion here is that a given map has an inverse on a sufficiently small neighborhood of any given point if and only if its differential has an inverse at the point. The statement of the implicit function theorem (Theorem 1.2) refers to a regular value of a map between Euclidean spaces. I remind you of the definition: Let U  Rm denote a given open subset and let c: U ! Rk denote a given smooth map. A point a 2 Rk is said to be a regular value of c if the matrix c* is surjective at all points in c1(a).

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Theorem 1.2 (the implicit function theorem) Fix nonnegative integers m  n. Suppose that U  Rm is an open set, c: U ! Rmn is a smooth map, and a 2 Rmn is a regular value of c. Then c1(a)  U has the structure of a smooth, n-dimensional manifold whose smooth structure is defined by coordinate charts of the following sort: Fix a point p 2 c1(a). Then there is a ball B  Rm centered at p such that the orthogonal projection from B to the kernel of c*jp restricts to c1(a) B as a coordinate chart. In addition, there is a diffeomorphism, j: B ! Rm such that j(B c1(a)) is a neighborhood of the origin in the n-dimensional linear subspace of points (x1, . . . , xm) with xnþ1 ¼ . . . ¼ xm ¼ 0. ¨

¨

This theorem is, of course, vacuous if m < n. By the way, this theorem is often used in conjunction with Theorem 1.3 (Sard’s theorem) Suppose that U  Rm is an open set, c: U ! Rn is a smooth map. Then the set of regular values of c have full measure. This means the following: Fix any ball in Rn. Delete the all-nonregular values from this ball. The volume of the resulting set is the same as that of the ball. Thus, a randomly chosen point from the ball has probability 1 of being a regular value. (A special case of Sard’s theorem (Theorem 14.14) is proved in Appendix 14.3.)

1.3

Submanifolds of Rm

Let f: Rm ! R denote a smooth function and let a 2 R denote a regular value of f. Then f1(a) is a smooth, (m1)-dimensional manifold; this is a consequence of the implicit function theorem. This is an example of an (m1)-dimensional submanifold of Rm. For a concrete example, take the function x ! f(x) ¼ jxj2 where jxj denotes the Euclidean norm of x. Any nonzero number is a regular value, but of course only positive regular values are relevant. In this case f1(r2) for r > 0 is the sphere in Rnþ1 of radius r. Spheres of different radii are diffeomorphic. Definition 1.4 A submanifold in Rm of dimension n < m is a subset, , with the following property: Let p denote any given point in . There is a ball Up  Rm around p and a map cp: Up ! Rmn with 0 as a regular value and such that  Up ¼ c1 p (0). ¨

4

The implicit function theorem says that any such  is a smooth, n-dimensional manifold. The following lemma can be used to obtain examples of submanifolds.

1.3

Submanifolds of Rm

Lemma 1.5 Suppose that n  m and that B  Rn is an open ball centered on the origin. Let j: B ! Rm denote a smooth, 1–1 map whose differential is everywhere injective. Let W  B denote any given open set with compact closure. Then j(W) is a submanifold of Rm such that jjW : W ! j(W) is a diffeomorphism. Proof of Lemma 1.5 This can be proved using the inverse function theorem (Theorem 1.1). To say more, fix any given point p 2 j(W). Needed is a smooth map, cp from a ball centered at p to Rmn that maps p to the origin, whose differential at p is surjective, and is such that j(W) near p is c1 p (0). To obtain this data, let z 2 W denote the point mapped by j to p. Let K  Rm denote the kernel of the adjoint of j*jp. This is an (m  n)-dimensional subspace because the differential of j at z is injective. Keeping this in mind, define l: W  K ! Rm by the rule l(x, v) ¼ j(x) þ v. Note that l(z, 0) ¼ p. As the differential of l at (z, 0) is an isomorphism, the inverse function theorem finds a ball, Up, of p and a smooth map Z: Up ! W  K such that the following is true: First, Z(Up) is an open neighborhood of (z, 0). Second, j  Z is the identity. Third, Z  j is the identity on Z(Up). Granted this, let p: W  K ! K denote the orthogonal projection. Identify K with Rnm and define the map cp: Up ! K ¼ Rnm by the rule cp(x) ¼ pZ(x). By construction, c1 p (0) ¼ j(Z(Up)) and cp jp is surjective. * Here is an example of a dimension 2 submanifold in R3. Fix r 2 (0, 1). The submanifold is the set of points (x1, x2, x3) 2 R3 where fðx1 ; x2 ; x3 Þ ¼ ððx21 þ x22 Þ1=2  1Þ2 þ x23 ¼ r2 : This is a torus of revolution; it is diffeomorphic to S1  S1 as can be seen by introducing an angle f 2 R /2pZ for the left-most S1 factor, a second angle j for the right-most factor, and then using these to parametrize the locus by the map that sends   ðf; jÞ ! x1 ¼ ð1 þ rcosfÞcosj; x2 ¼ ð1 þ rcosfÞsinj; x3 ¼ rsinf : A torus sitting in R3 looks like the surface of a tire inner tube of the sort that you would inflate for a bicycle. Figure 1.1 is a depiction viewed from above.

Figure 1.1

A torus sitting in R3

A surface in R3 with some number holes when viewed from above can also appear as a submanifold of R3. The notion of what constitutes a hole has

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rigorous definition; see for example Chapter 12 in the book Topology by James Munkres (2000). Chapter 14 in the present volume gives two equivalent definitions using differential geometric notions. Suffice it to say here that the torus depicted above has genus 1, and a standard pretzel has genus 2. What follows depicts a surface in R3 with a given genus g. This construction was 1 explained to the author by Curt McMullen. To start, fix r 2 (0; 100g ) and a 2 function h: R ! R with the following properties:    

h(x, y)  0 where jxj2 þ jyj2  1. k 2 For each k ¼ {1, . . . , g}, require h(x, y)  0 where jx  2g j þ jyj2  r2 . h(x, y) > 0 otherwise. @h j @h @x j þ j @y j 6¼ 0 at any point where h ¼ 0, thus on the circle of radius 1 about the n origin, and on the o g circles of radius r about the respective points in the k set x ¼ 2g ;y ¼ 0 . k¼1;:::;g

Figure 1.2 is a schematic of how h might look like in the case g ¼ 2. The black area is where h > 0; the white is where h  0. The corresponding surface in R3 is the set of points where the function ƒ ¼ z2  h(x, y) is zero. The condition on h in the fourth bullet guarantees that 0 is a regular value of ƒ and so ƒ1(0) is a smooth submanifold in R3. It is compact because h(x, y)  0 when x2 þ y2 is greater than 1. Any n-manifold can be realized as a submanifold of Rm for some m n. What follows is an argument for the case when the manifold in question is compact manifold. To start, fix a finite coordinate atlas for M. Label the elements in this atlas as {(U1, j1), . . . , (UN, jN)}. Fix a subordinate partition of unity, {wa}1aN. This is a set of smooth functions on M such that any given wa has support in the corresponding set Ua and such that 1aN wa ¼ 1 at each point. Appendix 1.2 gives a quick tutorial in such thing. Introduce the map

Figure 1.2

h in the case g¼2

1.4

Submanifolds of manifolds

c : M ! RNmþN ¼ 1aN ðR  Rm Þ that is defined by the following rule: Send any given point x 2 M to the point in the a’th factor of RN(mþ1) given by (1  wa(x), wa(x)fa(x)). The differential of this map is injective because there exists near any given point at least one index a such that fa 6¼ 0. If this is the case for a, then the map to Rmþ1 from a small neighbohood of this point given by x ! (1  wa(x), wa(x)ja(x)) is injective. This understood, the implicit function theorem can be used to infer that the image of c is a submanifold of RN(mþ1) and that c is a diffeomorphism onto its image.

1.3.1

Coordinates near a submanifold

The implicit function theorems can be used to construct “nice” coordinates for Rn near any given point of a submanifold. Here is what can be done: Let Y  Rn denote a submanifold of some dimension d, and let p denote any given point in Y. Then there is a neighborhood U  Rn of p and a diffeomorphism c: U ! Rn such that c(Y) near p is the locus of points (x1, . . . , xn) where xdþ1 ¼ xdþ2 ¼ . . . ¼ xn ¼ 0.

1.4

Submanifolds of manifolds

Suppose that M is a smooth manifold of dimension m. A subset Y  M is a submanifold of some given dimension n  m if the following is true: Fix any point p 2 Y and there is an open set U  M that contains p, and a coordinate chart map f: U ! Rm such that f(Y U) is an n-dimensional submanifold of Rm. Note that if c: M ! N is smooth, and if Y  M is a smooth submanifold, then the restriction cjY : Y ! N is smooth. ¨

1.4.1

Immersions and submersions

The notion of a submanifold is closely related to the notions of immersion and submersion. To say more, let M denote a manifold of dimension m and let n denote one of dimension Y. Suppose first that n  m. A smooth map c: Y ! M is said to be an immersion when it has the following property: Fix any point p 2 Y and an open set V  Y that contains p with a coordinate map fV : V ! Rn. Let U  M denote an open set containing c(p) with a coordinate map fU: U ! Rm. Then the differential of fU  c  fV1 at fV(p) is an injective linear map from Rn to Rm. This implies that c(V) is a n-dimensional submanifold of M on some small neighborhood of c(p).

7

1 : Smooth manifolds

Suppose next that n  m. The map c is said to be a submersion when the following is true: Fix p, V, and U as above. Then the differential of fU  c  f1 V at p is a surjective linear map from Rn to Rm. An application of the inverse function theorem leads to the conclusion that c1(c(p)) is a (nm)-dimensional submanifold of Y near p.

1.5

More constructions of manifolds

What follows are some simple ways to obtain new manifolds from old ones.

1.5.1

Products of smooth manifolds

The product of two circles has the following generalization: Suppose that M, N are smooth manifolds. Then M  N has a canonical smooth structure. A coordinate atlas for the latter can be taken as follows: Fix coordinate atlases, U for M and V for N, that define the respective smooth structures. Then a coordinate atlas for M  N is the collection {(U  V, j  c)}(U,j)2U, (V,c)2V. The dimension of M N is the sum of the dimensions of M and N. For example, the circle is S1; the n-fold product of the circle is called the n-dimensional torus.

1.5.2

Open subsets of manifolds

Let M denote a smooth manifold and W  M any open subset. Then W inherits the structure smooth manifold. Indeed, fix a coordinate atlas for M that defines the smooth structure. Then a coordinate atlas for the smooth structure on W can be taken to be the collection {(U W, j)}(U, j)2U . ¨

8

1.5.3

Quotients that are manifolds

Let M denote a smooth manifold. An equivalence relation on M is a collection of disjoint subsets in M whose union is the whole of M. I will use M/ to denote this collection of sets. An element in the set M/ is said to be an equivalence class; and when x and x0 are points in M, I write x x0 when they lie in the same equivalence class. A map p: M ! M/ is defined by sending any given point to its equivalence class. The set M/ is given the quotient topology whereby a set U  M/ is declared to be an open set if and only if p1(U) is open in M. It is often the case that M / inherits from M the structure of a smooth manifold. In particular, this occurs when the following is true: Let x 2 M.

1.6

More smooth manifolds: The Grassmannians

Then there is an open neighborhood V  M of x such that pjV is a 1–1 and a homeomorphism from V to p(V). If this is the case, then M / has a canonical smooth structure, this defined as follows: Let U denote a given coordinate atlas for M that defines the smooth structure. Fix an open cover, V, of M such that p is 1–1 on any set from V. A coordinate atlas for M in U’s equivalence class is defined by the collection {(U V, jjU V)}(U,j)2U, V2V. With this understood, a coordinate atlas that defines the smooth structure for M/ is given by the collection {(p(U V), j  p1: p(U V) ! Rn)}(U,j)2U, V2V . Note in particular that the map p from M to M / is a smooth map. Here is a first example: The circle S1 can be viewed as R / where the equivalence relation identifies points x and x0 if and only if x ¼ x0 þ 2pk for some k 2 Z. With S1 viewed as the unit circle in R2, the map p becomes the map sending t 2 R to the point (cos(t), sin(t)). When I think of S1 in terms of this equivalence relation, I will write it as R /2pZ. By the same token, the n-dimensional torus n S1 can be viewed as Rn/

where the equivalence relation identifies points x and x0 if each entry of x differs from the corresponding entry of x0 by 2p times an integer. Here is a second example: The real projective space RPn ¼ Sn/ where the equivalence relation identifies unit vectors x and x0 in Rnþ1 if and only if x ¼  x0 . As a parenthetical remark, note that RPn can be viewed as the set of 1-dimensional linear subspaces in Rnþ1. The correspondence is as follows: A given 1-dimensional subspace has precisely two vectors of unit length, one is 1 times the other. These two vectors define a point in RPn. Conversely, a point in RPn gives the linear subspace that is spanned by either of its corresponding points in Sn. For the third example, fix two integers n  2 and m  1. Define an equivalence relation on S2n1 as follows: The point x ¼ (x1, x2, . . . , x2n) is 0 equivalent to x0 ¼ (x10 , x20 , . . . , x2n ) if and only if for each k 2 {1, . . . , n}, the following is true: ¨

¨

¨

¨

x2kþ10 ¼ cosð2pj=mÞx2kþ1  sinð2pj=mÞx2k ; x2k0 ¼ cosð2pj=mÞx2k þ sinð2pj=mÞx2kþ1 with j some integer. Note that RP2nþ1 is obtained when m ¼ 2. Manifolds of this sort are examples of lens spaces.

1.6

More smooth manifolds: The Grassmannians

As noted above, the real projective space RPn can be viewed as the set of 1-dimensional vector subspaces in Rnþ1. In general, if m > n, then the set of n-dimensional vector subspaces in Rm also has the structure of a smooth

9

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1 : Smooth manifolds

manifold. The latter manifold is denoted by Gr(m; n) and is called the Grassmannian of n-planes in Rm. The manifold structure for Gr(m; n) can be obtained by exhibiting a coordinate atlas whose transition functions are diffeomorphisms. The first task, however, is to give Gr(m; n) a suitable topology. The latter is the metric topology defined as follows: Let V  Rm denote a given n-dimensional subspace. Let V : Rm ! V denote the orthogonal projection. An open neighborhood of V is the set, O V, of subspaces V0 such that V : V0 ! V is an isomorphism. A basis of open neighborhoods of V is indexed by the positive numbers; and the e > 0 member, O V,e, consists of the n-dimensional subspaces V0  Rm such that jV(v)  vj < ejvj for all v 6¼ 0 in V0. As a manifold, Gr(m; n) has dimension n(mn). The neighborhood O V will serve as a coordinate chart centered around V; thus a map from O V to Rn(nm) that gives the local coordinates is needed. In order to give the coordinate map, first introduce M(m, n) to denote the vector space of (m  n) m matrices. The entries of any given matrix p 2 M(m, n) serve as coordinates and identify this space with Rn(mn). Now, let V  Rm denote a given n-dimensional subspace and let V⊥  Rm denote its orthogonal complement. Use V⊥: Rm ! V⊥ in what follows to denote the orthogonal projection to V⊥. Fix a basis for V to identify it with Rm and fix a basis for V⊥ to identify it with Rmn. Use these bases to identify the space of linear maps from V to V⊥ (denoted by Hom(V; V⊥)) with Rn(mn). Now, write Rm ¼ V  V⊥. Suppose that V0 2 O V. Since V : V0 ! V is an isomorphism, there is a linear inverse LV,V : V ! V0. This understood, the composition V⊥  LV,V0 is a linear map from V to V⊥, thus an element in Hom(V; V0 ) ¼ Rn(mn). This point is defined to be jV(V0 ). It is an exercise to check that the map jV is continuous. To see that it is a homeomorphism, it is enough to exhibit a continuous inverse: The inverse sends a given n  (mn) matrix p 2 Hom(V; V⊥) ¼ Rn(mn) to the m-dimensional linear subspace j1 V (p) ¼ {(v, pv) : v 2 V}. It remains yet to verify that the transition function between any two intersecting charts defines a smooth map between domains in Rn(mn). This task is deferred to Appendix 2.1. It is left as a linear algebra exercise to verify that the transition functions between two charts O V and O V0 are diffeomorphisms. I also leave it as an exercise to verify that the smooth structure just defined for Gr(nþ1; 1) is the same as that given previously for RPn.

Appendix 1.1

Appendix 1.1

How to prove the inverse function and implicit function theorems

How to prove the inverse function and implicit function theorems

What follows are detailed outlines of proofs first of the inverse function theorem and then of the implicit function theorem. See, for example, Differential Topology by Victor Guillemin and Alan Pollack (1974) for unabridged proofs.

A1.1.1

Proof of the inverse function theorem (Theorem 1.1)

The first point to make is that no generality is lost by taking p and c(p) to be the origin. Let m denote the n  n matrix c*j0. I look for an inverse, s, that has the form sðxÞ ¼ m1 x þ fðxÞ where f is such that jf(x)j  c0 jxj2. Here I have introduced a convention that is used in this chapter and all subsequent chapters: What is denoted by c0 is always a constant, greater than 1, whose value increases on subsequent appearances. The use of this generic notation is meant to avoid the proliferation of different symbols or subscripts to denote positive numbers which are either greater than 1 or less than 1, but are such that the precise value is of no real consequence to the discussion at hand. The convention here takes c0 to be greater than 1, so when it is important that a positive number have value less than 1, I use c01. To find f, I use Taylor’s theorem with remainder to write c(x) ¼ mx þ r(x) where r obeys jr(x)j  c0jxj2. Note also that any given partial derivative of r obeys j@ jr(x)j  c0jxj. In any event, the map s is an inverse to c if an only if c(s(x)) ¼ x for x near 0 in Rn. In terms of f, this demands that m fðxÞ þ rðm1 x þ fðxÞÞ ¼ 0: This is to say that fðxÞ ¼ m1 rðm1 x þ fðxÞÞ: This sort of equation can be solved for any given x using the contraction mapping theorem (Theorem 1.6). The point being that the desired solution, f(x), to the preceding equation is a fixed point of the map from Rn to Rn that sends y to T x(y) ¼ m1r(m1x þ y). The contraction mapping theorem asserts the following:

11

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1 : Smooth manifolds

Theorem 1.6 (the contraction mapping theorem) Let B  Rn denote a closed ball and T a map from B to itself. The map T has unique fixed point in B if jT (y)  T (y0 )j  (1  d) jy  y0 j for some positive d and for all y, y0 2 B. Proof of Theorem 1.6 Start with y0 2 B and then construct a sequence {yk ¼ T (yk1)}k¼1,2, . . . . This is a Cauchy sequence because jyk  yk1j ¼ jT (yk1)  T (yk2)j  (1  d)jyk1  yk2j. Let yf denote the limit point. This point is in B and T (yf) ¼ yf. There can’t be any other fixed point in B; were y0 such a point, then jyf  y0 j ¼ jT (yf)  T (y0 )j  (1  d)jyf  y0 j which can be true only if yf ¼ y0. In the case at hand, jT x(y)j  c0(jxj2 þ jyj2) because r comes from the remainder term in the Taylor’s expansion for c. This implies that T x maps any ball of 1 0 radius r < 12 c1 0 to itself if jxj  c0 r. For the same reason, jT x(y)  T x(y )j 0  c0(jxj þ jyj) jy  y j, and so T x is a contraction mapping on such a ball if jxj  2 c1 0 r. The unique fixed point is f(x) that obeys jf(x)j  c0r . It remains yet to prove that the map x ! f(x) is smooth. A proof that such is the case can also be had by successively using the contraction mapping theorem to analyze first the difference jf(x)  f(x0 )j, and then successively higher difference quotients. For the first difference quotient, use the fact that f(x) and f(x0 ) are respective fixed points of T x and T x0 to write their difference as fðxÞ  fðx0 Þ ¼ m1 ðrðm1 x þ fðxÞÞÞ  rðm1 x0 þ fðx0 ÞÞ Given that r is the remainder term in Taylor’s theorem, and given that x and x0 lie in the ball of radius r, and that f(x) and f(x0 ) lie in the ball of radius c0r2, this last equation implies that jf(x)  f(x0 )j  c0r (jx  x0 j þ jf(x)  f(x0 )j). 1 0 0 In particular, if r < 12 c1 0 , this tells us that jf(x)  f(x )j  c0 jx  x j.

A1.1.2

Proof of the implicit function theorem (Theorem 1.2)

What follows is a sketch of the proof of the implicit function theorem. The first step constructs a local coordinate chart centered on any given point in c1(a). Let p denote such a point. Introduce K  Rm to denote the kernel of c*jp, and let p: Rm ! K denote the orthogonal projection. Now define a map, j: Rm ! Rm ¼ K  Rmn by sending any given point x 2 Rm to (p(x  p), c(x)  a). The differential of this map at p is an isomorphism, and so the inverse function theorem finds balls BK  K and B0  Rmn about the respective origins with a map s: BK  B0 ! Rm that obeys s(0) ¼ p and is such that j  s ¼ identity. In particular, this has the following implication: Let y 2 BK. Then c(s(y, 0)) ¼ a. As a consequence, the map from BK to Rm that sends y to s(y, 0) is 1–1 and invertible. The inverse is the map x ! p(x  p) from Rm to K ¼ Rn, and so the latter map gives local coordinates for c1(a) near p.

Appendix 1.2

Partitions of unity

Granted the preceding, it remains yet to prove that the transition functions for these coordinate charts are smooth and have smooth inverses. There is nothing terribly complicated about this, and so it is left to the reader.

Appendix 1.2

Partitions of unity

This appendix says something about how to construct a locally finite partition of unity for a coordinate atlas. The story starts on [0, 1). What follows directly describes a smooth function that is zero at points t 2 [0, 1) with t  1 and is nonzero at all points t 2 [0, 1) with t < 1. The function, w, is defined by the rule whereby  w(t) ¼ e1/(1t)  w(t) ¼ 0

for t < 1. for t  1.

Granted this, suppose that U is a locally finite coordinate atlas for a given smooth, n-dimensional manifold with 0 2 Rn in all chart images. Since U is locally finite, one can assign to each (U, jU) 2 U a positive real number, rU, such that the following is true: Let Ur denote the inverse image via jU of the ball of radius rU about the origin in Rn. Then the collection {Ur}(U,j)2U is an open cover of M. This understood, associate to any given (U, jU) 2 U the function sU on M that is defined as follows: If p is not in U, then sU(p) ¼ 0. If p is in U, then sU(p) is equal to w(jjU(p)j/rU). Note that sU(p) is nonzero on Ur but zero on the complement of U. The collection {sU}(U,j)2U is almost, but not quite, the desired subordinate P partition of unity function. It is certainly the case that (U,j)2U sU is nowhere zero, but this sum is not necessarily equal to 1. The desired subordinate P partition of unity consists of the set {wU}(U,j)2U where wU ¼ ( (U0 ,j0 )2U sU0 )1sU.

Additional reading  Differential Topology, Victor Guillemin and Alan Pollack, Prentice-Hall, 1974.  An Introduction to Manifolds, Loring Tu, Springer, 2008.  Elementary Differential Topology, James R. Munkres, Princeton University Press, 1966.  An Introduction to Differentiable Manifolds and Riemannian Geometry, William M. Boothby, Academic Press, 1975.

13

2

Matrices and Lie groups

Matrices and especially invertible matrices play a central role in most all of differential geometry. In hindsight, I don’t find this to be surprising. The role stems ultimately from Taylor’s theorem: A smooth map between two Euclidean spaces can be well approximated near any given point by a linear map, this the differential at the point. The inverse and implicit function theorems (Theorems 1.1 and 1.2) depend entirely on this observation. In any event, matrices and notions from linear algebra appear in all of the subsequent chapters. Constructions involving matrices also give some interesting and explicit examples of smooth manifolds.

2.1

The general linear group

I use M(n; R) to denote the vector space of n  n matrices with real entries. This 2 is a copy of the Euclidean space Rn with the entries of a matrix giving the coordinate functions. Matrices m and m0 can be multiplied, and matrix multiplication gives a smooth map, (m, m0 ) ! mm0 , from M(n; R)  M(n; R) to M(n; R). This is smooth by virtue of the fact that the entries of mm0 are linear functions of the coordinates of m, and also linear in the coordinates of m0 . The space M(n; R) has two special functions. The first function, det: M(n; R) ! R sends any given matrix to its determinant. This is a smooth function since it is a polynomial of degree n in the entries. The determinant of a matix, m, is denoted by det(m).The other function of interest is the trace. The trace is the sum of the diagonal entries and so a linear function of the entries. The trace of a matrix m is denoted by tr(m). Let Gl(n; R) denote the subspace in M(n; R) of invertible matrices. This is to say that m 2 Gl(n; R) if and only if the det(m) 6¼ 0. This is an open subset of M(n; R), and so de facto Gl(n; R) is a smooth manifold of dimension n2. In particular, the n2 entries of the matrix restrict to Gl(n; R) near any given point to give local coordinates. The multiplication map restricts to Gl(n; R) as a smooth map.

2.2

Lie groups

The manifold Gl(n; R) has a canonical diffeomorphism, this the map to itself that sends any given matrix m to the inverse matrix m1. This map is smooth because any given entry of m1 is the quotient by det(m) of a polynomial in the entries of m. For example, in the 2  2 case, write   a b m¼ ; c d and then its determinant is ad  bc and its inverse is   1 d b 1 : m ¼ a ad  bc c The form just asserted for the entries of the inverse in the n > 2 case (a polynomial divided by the determinant) can be seen by using the reduced row echelon form algorithm to construct the inverse. What follows gives a very useful formula for the directional derivatives of the map m ! m1. Let a 2 M(n; R) denote any given matrix. Then the directional derivatives of the coordinates of map m ! m1 in the direction a are the entries of the matrix m1am1. Consider, for example, the coordinate given by the (i,j) entry, (m1)ij: The directional derivative in the direction a of this function on Gl(n; R) is (m1am1)ij. In particular, the partial derivative of the function m ! (m1)ij with respect to the coordinate mrs is (m1)ir(m1)sj.

2.2

Lie groups

A group is a set with a fiducial point and two special maps. To say more let G denote the set. The fiducial point is denoted in what follows by i; it is called the identity. The first of the maps is the multiplication map, this a map from G  G to G. It is customary to denote this map by (g, g0 ) ! gg0 . This map is constrained so that gi ¼ g and also ig ¼ g. The other map sends G to G; it is the inverse. It is customary to write the inverse as g ! g1. The inverse map is constrained so that gg1 ¼ i ¼ g1g. A Lie group is a group with the structure of a smooth manifold such that both the multiplication map and the inverse map are smooth. The manifold Gl(n; R) as just described is the Ur example of a Lie group. Note that Rn with the origin playing the role of i, with addition playing the role of multiplication, and with the map x ! x playing the role of inverse is a group. The latter is an example of an Abelian group, this a group with the property that gg0 ¼ g0g for all pairs g and g0 .

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2 : Matrices and Lie groups

It is often the case that interesting groups occur as subgroups of larger groups. By way of reminder, a subgroup of a group G is a subset, H, that contains i, is mapped to itself by the inverse map, and is such that multiplication maps H  H to H. Lemma 2.1 A subgroup of a Lie group that is also a submanifold is a Lie group with respect to the induced smooth structure. Proof of Lemma 2.1 This follows by virtue of the fact that the restriction to a submanifold of any smooth map to any given manifold defines a smooth map from the submanifold to the given manifold.

2.3

Examples of Lie groups

The group Gl(n; R) contains sub-Lie groups that arise most often in differential geometric constructions.

2.3.1

The group Sl(n; R)

Let Sl(n; R)  Gl(n; R) denote the subset of matrices whose determinant is 1. This is a subgroup by virtue of the fact that det(mm0 ) ¼ det(m)det(m0 ). It is a smooth submanifold of dimension n2  1. This follows from the fact that the differential of this function can be written as dðdetÞjm ¼ detðmÞ ∑i;j ðm1 Þij dmij ¼ detðmÞ trðm1 dmÞ: Here, dm denotes the n  n matrix whose (i, j)’th entry is the coordinate differential dmij. As can be seen from the preceding formula, the differential of det is not zero where det is nonzero. As it turns out, the group Sl(2; R) is diffeomorphic to S1  R2. This can be seen by using a linear change of coordinates on M(2; R) that writes the entries in terms of coordinates (x, y, u, v) as follows: m11 ¼ x  u; m22 ¼ x þ u; m12 ¼ v  y and m21 ¼ v þ y: The condition that det(m) ¼ 1 now says that x2 þ y2 ¼ 1 þ u2 þ v2. This understood, the diffeomorphism from S1  R2 to Sl(2; R) sends a triple (y, a, b) to the matrix determined by x ¼ (a2 þ b2)1/2 cos(y), y ¼ (a2 þ b2)1/2 siny, u ¼ a, v ¼ b. Here, y 2 [0, 2p] is the angular coordinate for S1.

2.4

2.3.2

Some complex Lie groups

The orthogonal groups O(n) and SO(n)

The orthogonal group O(n) is the set of matrices m 2 Gl(n; R) such that mTm ¼ 1. Here, (mT)ik ¼ mki. Since (ab)T ¼ bTaT, the set of orthogonal matrices forms a subgroup. The group SO(n) sits in O(n) as the subgroup of orthogonal matrices with determinant 1. To see that SO(n) and O(n) are manifolds, introduce Sym(n; R) to denote the vector subspace of n  n symmetric matrices. This is to say that a matrix h is in Sym(n; R) when hT ¼ h; thus when hij ¼ hji. This vector space has dimension n(nþ1)/2. The vector space Sym(n; R) is a linear subspace of the Euclidean space M(n; R), and so a version of Rn(nþ1)/2. Let c: M(n; R) ! Sym(n; R) denote the map m ! mT m. This map is quadratic in the entries of the matrix, and so a smooth map between Euclidean spaces. As is explained in the next paragraph, the identity matrix i 2 Sym(n; R) is a regular value of c. This being the case, it follows from the implicit function theorem that c1(i) is a submanifold of Gl(n; R) and so a Lie group. To see that i is a regular value, one must verify that the differential of i at any given matrix m 2 O(n) is surjective. To see this, note that the differential of c at a given matrix m can be written as c*jm ¼ dmTm þ mTdm, where dm is the matrix of coordinate differentials with (i, j) entry equal to dmij. To explain this notation, I should tell you how to interpret dmTm þ mTdm as a linear map from the Euclidean space M(n; R) to the Euclidean space Sym(n; R). This linear map acts as follows: Let a 2 M(n; R) denote any given matrix. Then c jm(a) is the * symmetric matrix aTm þ mTa. The rule here is to replace dm with a. Note that this notation is used throught this book, so it is best to come to terms with it. (It was used above to write d(det)jm as det(m) tr(m1dm).) In any event, granted this identification of c jm it is necessary to prove the following: Given a * symmetric matrix h, there exists a matrix a such that aTm þ mTa ¼ h. Given that mmT ¼ i, this equation is solved by taking a ¼ 12 m h. Note that SO(3) is diffeomorphic to the quotient space RP3 ¼ S3/ where

equates any given x with  x. Meanwhile, SO(4) is diffeomorphic to (S3  S3)/

where the equivalence relation equates a pair (x, y) 2 SO(3) only with itself and with (x, y).

2.4

Some complex Lie groups

The group Gl(n; R) has as its analog the group Gl(n; C) of n  n invertible matrices with complex number entries. There are two equivalent ways to view Gl(n; C) as a Lie group. The first deals with complex numbers right off the bat by introducing the vector space (over C) of n  n complex matrices. The latter is denoted by M(n; C). This vector space can be viewed as a version of

17

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2 : Matrices and Lie groups 2

R2n by using the real and imaginary parts of the n2 entries as coordinates. Viewed in this light, multiplication of matrices defines a smooth map from M(n; C)  M(n; C) to M(n; C) by virtue of the fact that the entries of a product mm0 is linear in the entries of m and linear in those of m0 . The determinant function maps M(n; C) to C, and this is a smooth map since it is a polynomial of degree n in the entries. The latter map is also denoted by det(·), with it understood that this version has values in C while the version defined on the space of matrices with real entries has values in R. The group Gl(n; C) appears as the open set in the Euclidean space M(n; C) where the determinant is nonzero. The matrix multiplication for M(n; C) restricts to a smooth map from 2 Gl(n; C) to Gl(n; C). Meanwhile, the inverse m ! m1 is smooth because any given entry of m1 is the quotient of an n’th order polynomial in the entries of m by the nonzero, smooth, C-valued function given by the determinant function det(·). The second and equivalent way to view Gl(n; C) as a manifold does not introduce complex numbers until late in the game. This view is ultimately useful because it introduces some constructions that play a role in later chapters. This approach views Gl(n; C) as a subgroup in Gl(2n; R). To see Gl(n; C) in this light, I first introduce the notion of an almost complex structure, this an element j 2 M(2n; R) with j2 ¼ i with i again denoting the identity matrix. The canonical example is j0 whose n ¼ 1 and n ¼ 2 versions are 0 1   0 1 0 0 0 1 B 1 0 0 0C C: ;B @ 1 0 0 0 0 1 A 0 0 1 0 In the n > 2 case, j0 is given as follows: Let ek denote the basis vector for R2n that has 1 in the entry k and zero in all other entries. Then j0 is defined by the rules j0 e2n1 ¼ e2n and j0e2n ¼ e2n1. Use Mj for the moment to denote the vector subspace of M(2n; R) of matrices m such that mj0  j0m ¼ 0. This is to say that the entries of m are such that m2k;2i ¼ m2k1;2j1 and m2k;2j1 ¼ m2k1;2j for each pair k, j 2 {1, . . . , n}. As is explained momentarily, the vector space Mj is M(n; C). In any event, Mj is a vector space over R, and so a Euclidean space. Introduce for the moment Gj  Mj to denote the subset of invertible elements. Note that Gj is not empty because the identity matrix i and also j0 are in Gj. As an open subset of a Euclidean space, Gj is a smooth manifold. It is also a group by virtue of the fact that mm0 j0 ¼ mj0m0 ¼ j0mm0 when m and m0 are both in Gj. Thus it is a Lie group. I shall explain in a moment how to identify Gj with Gl(n; C).

2.5

The groups Sl(n; C), U(n) and SU(n)

To see that Mj is M(n; C), note first that the eigenvalues of j0 are i, and any matrix that commutes with j0 must preserve the eigenspaces. Indeed, let {e1, . . . , e2n} denote the standard basis for R2n, and then a basis of eigenvectors for the eigenvalue i are {v1 ¼ e1  ie2, . . . , vn ¼ e2n1  ie2n}. The corresponding basis of eigenvectors for the eigenvalue i is given by the complex conjugates of {v1, . . . , vn}. If m commutes with j0, then it must act as m(vk) ¼ akjvj (note the summation convention for repeated indices) where akj 2 C. Likewise, m(v k) ¼ akj vj . Here and in what follows, the overscore indicates complex conjugation. Note that the number akj 2 C can be written in terms of the entries of m as akj ¼ m2k1,2j1 þ im2k,2j1. The preceding implies that m is determined by n2 complex numbers and so the dimension of Mj is 2n2. The identification between Mj and M(n; C) comes about by associating to any given m 2 Mj the n  n matrix with complex entries given by the numbers {akj}. The identification between Mj and M(n; C) identifies the subset Gj  Mj with the group Gl(n; C). To elaborate, suppose that a is a given element in Mj and let mC denote the corresponding element in M(n; C). As an element in M(2n; R), the matrix m has a real valued determinant; and as an element in M(n; C), the matrix mC has a C-valued determinant. These two notions of a determinant are such that detðmÞ ¼ jdetðmC Þj2 : Thus, the real 2n  2n matrix a is an invertible element in Mj if and only if the corresponding complex, n  n matrix mC is an invertible element in M(n; C). This correspondence between Gj and Gl(n; C) has the following property: If m0 and m0C are paired, then the correspondence pairs mm0 with mCm0C. As a consequence, if the correspondence pairs m 2 Gj with mC 2 Gl(n; C), it then pairs m1 with mC1.

2.5

The groups Sl(n; C), U(n) and SU(n)

The three most commonly seen subgroups of Gl(n; C) are the subject of this part of the chapter.

2.5.1

The group Sl(n; C)

This is the subset of matrices in M(n; C) with determinant 1. To be precise, the determinant here is that of an n  n matrix with complex entries. For example, if the matrix is an n  n diagonal matrix, the determinant is the product of the diagonal entries. In particular, the determinant function on M(n; C) is to be viewed as a map, det: M(n; C) ! C.

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As a parenthetical remark, when M(n; C) is viewed as the subvector space Mj  M(2n; R), it inherits a determinant function from M(2n; R), this mapping to R. Let me call the latter detR for the moment, and call the aforementioned determinant function detC. These two functions on Mj are related by the rule detR ¼ jdetCj2. This can be seen as follows: A matrix m 2 M(n; C) has n complex eigenvalues {l1, . . . , ln}. When viewed as an element in Mj and thus in M(2n; R), this matrix has 2n eigenvalues. The first n are {l1, . . . , ln}, and the remaining n eigenvalues are the latter’s complex conjugates. In any event, to show that Sl(n; C) is a submanifold, I should prove that the complex determinant det: M(n; C) ! C has 1 as a regular value. Here is how to do this: The directional derivative of det at an invertible matrix m 2 M(n; C) in the direction of a given matrix a 2 M(n; C) is det(m) tr(m1a). To see that any given value in C can be had by choosing an appropriate matrix a, let c 2 C denote the desired value, and then take a ¼ c det(m)1 m.

2.5.2

The group U(n)

This is the subset of matrices m 2 M(n; C) that obey m{m ¼ i . This is to say that U(n) is the subgroup of matrices m with m1 ¼ m{. Here, m{ is the matrix whose (i, j) entry is the complex conjugate of the (j, i) entry of m. To prove that U(n) is a Lie group, I need only verify that it is a submanifold on M(n; C). To start, introduce the vector space Herm(n) of n  n matrices h that obey h{ ¼ h. I remind you that such a matrix is said to be Hermitian. This is a vector space over C of dimension 12 n(n þ 1); its complex coordinates can be taken to be the entries on and above the diagonal. Define a map c: M(n; C) ! Herm(n) by the rule c(m) ¼ m{m. Since U(n) ¼ c1(i), this subset is a submanifold if the identity matrix is a regular value of c. To see that such is the case, I note that the differential c at a matrix m sends a given matrix a to c jm(a) ¼ m{a þ a{m. The * * differential c jm is surjective if and only if any given h 2 Herm(n) can be written * as m{a þ a{m for some a 2 M(n; C). Given that m is invertible, then so is m{; its inverse is (m1){. This understood, take a ¼ 12 (m{)1h.

2.5.3

The group SU(n)

This is the subgroup in U(n) of matrices m with det(m) ¼ 1. Note that if m 2 U(n), then jdet(m)j ¼ 1, since the equation m{m ¼ i implies that det(m{m) ¼ det(i) ¼ 1, and so det(m{) det(m) ¼ 1. Given that the determinant of m{ is the complex conjugate of the determinant of m, this implies that jdet(m)j2 ¼ 1. To see that SU(n) is a Lie group, it is sufficient to prove that it is a submanifold of M(n; C).

2.6

Notation with regards to matrices and differentials

For this purpose, define a map f: M(n; C) ! Herm(n)  R by the rule f(m) ¼ (m{m, 2i (det(m)  det(m{)). Noting that det(m{) is the complex conjugate of det(m), it follows that SU(n) is the component of f1((i, 0)) that consists of matrices with determinant equal to 1. The other component consists of matrices with determinant equal to 1. To see that f is surjective, I must verify that its differential at any given m 2 SU(n) is surjective. The latter at a matrix m sends any given matrix a 2 M(n; C) to the pair in Herm(n)  R whose first component is m{a þ a{m and whose second component is the imaginary part of det(m) tr(m1a). Let (h, t) denote a given element in Herm(n)  R. Given that m 2 SU(2), then a ¼ 12 mh  i tm is mapped by f jm to (h, t). *

2.6

Notation with regards to matrices and differentials

Suppose that f is a function on M(n; R). Since the entries of a matrix serve as the coordinates, the differential of f can be written as  @  X f dmij : df ¼ 1i;jn @m ij I remind you that this differential form notation says the following: The directional derivative of f in the direction of a matrix a is  @  X f aij : 1i;jn @m ij @ @ The latter can be written compactly as tr ð@m f aT Þ if it is agreed to introduce @m f T @ to denote the matrix whose (i, j) component is @mij f . Here, (·) denotes the transpose of the indicated matrix. The preceding observation suggests viewing the collection of coordinate differentials {dmij} as the entries of a matrix of differentials, this denoted by dm. Introducing this notation allows us to write dƒ above as ! ! @ T f dm tr : @m

Here, dmT is the matrix of differentials whose (i, j) entry is dmji. This notation is commonly used, and in particular, it will be used in subsequent parts of this book. An analogous short-hand is used when f is a function on M(n; C). In this case, the entries of a matrix serve as the complex coordinates for M(n; C). This is to say that their real and imaginary parts give M(n; C) its Euclidean coordinates as a manifold. The differential of the function f can be written concisely as

21

2 : Matrices and Lie groups

 @    @   f dmT þ tr f dm{ : tr  @m @m Note that this notation makes sense even when f is complex valued. For that matter, it makes sense when f is replaced by a map, c, from either M(n; R) or M(n; C) to a given vector space, V. In the case of M(n; R), the differential of c can be written succinctly as  @   c dmT : C* ¼ tr @m To disentangle all of this, suppose that V has dimension q, and one is given a basis with respect to which c has coordinates (c1, . . . , cq). Then what is written above says the following: The differential of c at a given matrix m is the linear map from M(n; R) to V that sends a matrix a to the vector with coordinates X  X @ 1 @ q : c a ; : : : ; c a ij ij 1i;jn @m 1i;jn @m ij ij There is, of course, the analogous notation for the case when c maps M(n; C) to a vector space.

Appendix 2.1

The transition functions for the Grassmannians

This appendix ties up a loose end from Chapter 1.6; it supplies a proof that the transition functions between any two coordinate charts in a given Grassmannian is a smooth map. To set the stage, fix integers m > 1 and n 2 {1, . . . , m} so to define the Grassmannian Gr(m; n) as described in Chapter 1.6. Let V and V0 denote two elements in Gr(m; n) and let O V and O V0 denote the respective coordinate charts from Chapter 1.6. Coordinate maps jV1: Rn(mn) ! O V 0 0 and jV1 : Rn(mn) ! O V are described. What follows explains why jV0  jV1 is smooth on its domain of definition. To start this task, introduce {ei}1in and {ua}1amn to denote the respective orthonormal bases chosen for Vand V⊥. Use {e0 i}1in and {u0 a}1amn to denote their respective V0 counterparts. Now let W  O V O V0 and let p 2 Rn(mn) and p0 2 Rn(mn) denote respective images under jV and jV0. Identify Rn(mn) with the vector space of matrices with n rows and m columns so as to write the components 0 of p as {pia}1in,1amn and those of p0 as {pia }1in,1amn. By definition, the coordinate p ¼ jV(W) is such that any given w 2 W can be parametrized by Rn as   X X v e þ p u w¼ a i i ia 1in 1amn ¨

22

Appendix 2.1

The transition functions for the Grassmannians

with v 2 Rn. Likewise, the coordinate p0 ¼ jV0 (W) is defined so that   X X 0 0 0 0 w¼ v e þ p u i 1in i 1amn ia a with v0 2 Rn. Note in particular that any given w 2 W is parametrized in two ways by points in Rn. This being the case, the two vectors v and v0 must be functionally related. To be explicit, the i’th component of v0 can be written in terms of v by taking the inner product of ei0 with w and using the preceding equations to first identify the latter with vi0, and then using the other equation above to write this inner product in terms of v. The result of equating these two expressions for ei0·w is   X X 0 0 v e e þ p e u v0i ¼ i k 1kn k 1amn ka j a where x·x0 denotes the Euclidean inner product on Rm. This last equation P asserts that vi0 ¼ 1kn Gikvk where the numbers {Gik}1i,kn are Gik ¼ e0i ek þ ∑1an pka e0i ek : These numbers can be viewed as the entries of an n  n matrix G. Note that this matrix G is invertible when the respective projections V and V0 map W isomorphically to V and V0. This is to say that G 2 Gl(n; R) when W 2 OV O V0. The assignment of p 2 Rn(mn) to G defines a smooth map from jV(O V O V0 ) in Rn(mn) into Gl(n; R) since the expression above for G’s entries define an affine function of the coordinate functions on Rm(nm). Save the matrix G for the moment. Use the expression that parametrizes w 2 W in terms of v0 2 Rn to write ua0 ·w in terms of v0 as X v0 p0 : 1in i ia ¨

¨

Meanwhile, ua0 ·w can also be written in terms of v using the expression that writes w in terms of v:   X X 0 0 v u e þ p u u : k k b a 1in 1bmn kb a Keeping in mind that these two expressions are equal, take v0 to be i’th basis vector in Rn to see that   X X 1 0 0 p0ia ¼ ðG Þ u e þ p u u : k b a ik 1kn 1bmn kb a Here, G1 is the inverse of the matrix G. With the entries of G viewed as a function of p 2 jV(O V O V0 ), the latter expression writes p0 as a function of p. This function is the transition function jV0  jV1. This expression for p0 as a

23

¨

2 : Matrices and Lie groups

function on jV(O V O V0 ) defines a smooth function on jV(O V the inverse map from Gl(n; R) to itself is a smooth map.

¨

¨

24

O V0 ) because

Additional reading  Matrix Groups, An Introduction to Lie Group Theory, Andrew Baker, Springer, 2002.  Lie Groups, An Introduction through Linear Groups, Wulf Rossmann, Oxford University Press, 2002.  Matrix Groups for Undergraduates, Kristopher Tapp, American Mathematical Society, 2006.  Lie Groups, Beyond an Introduction, Anthony W. Knapp, Birkhauser Boston, 1966.  Differential Geometry, Lie Groups and Symmetric Spaces, Sigurdur Helgason, American Mathematical Society, 1978.  Lie Groups, Lie Algebras and Representations, An Elementary Introduction, Brian C. Hall, Springer, 2003.

3

Introduction to vector bundles

Euclidean space is a vector space, but the typical manifold is not. Vector bundles bring vector spaces back into the story. To a first approximation, a vector bundle (over a given manifold) is a family of vector spaces that is parametrized by the points in the manifold. Some sort of vector bundle appears in most every aspect of differential geometry. Chapter 3 introduces these objects and gives the first few examples. By way of a warning, I give my favorite definition first; the standard definition is given subsequently in Chapter 3.2. (These definitions describe the same objects.)

3.1

The definition

Let M denote a smooth manifold of dimension m. A real vector bundle over M of fiber dimension n is a smooth manifold, E, of dimension n þ m with the following additional structure:  There is a smooth map p: E ! M.  There is a smooth map oˆ: M ! E such that p  oˆ is the identity.  There a smooth map m: R  E ! E such that a) p(m(r, v)) ¼ p(v) b) m(r, m(r,0 v)) ¼ m(rr,0 v) c) m(1, v) ¼ v d) m(r, v) ¼ v for r 6¼ 1 if and only if v 2 im(oˆ).  Let p 2 M. There is a neighborhood, U  M, of p and a map lU: p1(U) ! Rn such that lU: p1(x) ! Rn for each x 2 U is a diffeomorphism obeying lU(m(r, v)) ¼ rlU(v). The first four remarks that follow concern notation. The remaining two remarks are more substantive.

26

3 : Introduction to vector bundles

Remark 3.1 The map from E to E given by v ! m(r, v) for any given r 2 R is called multiplication by r; this is because the map m defines an action on E of the multiplicative group of real numbers. This R-action is reflected in the notation used henceforth that writes m(r, v) as rv. Remark 3.2 If W  M is any given set, then p1(W)  E is denoted by EjW. The inverse images via p of the points in M are called the fibers of E; and if p 2 M is any given point, then the fiber p1(p) is said to be the fiber over p. The map p itself is called the bundle projection map, or just the projection map. Remark 3.3 The map oˆ is called the zero section. The image of oˆ is also called the zero section; but it is almost always the case that these two meanings are distinguished by the context. This terminology reflects the fact that item (d) of the third bullet and the fourth bullet require lU  oˆ ¼ 0 2 Rn. Remark 3.4 The map lU defines a diffeomorphism, jU : EjU ! U  Rn given by jU(v) ¼ (p(v), lU(v)). It is customary to talk about jU rather than lU. The map jU is said to be a local trivialization for E. Remark 3.5 Let U  M denote an open set where there is a map lU: EjU ! Rn as above. The map lU can be used to give a local coordinate chart map for E in the following way: Let (W, j: W ! Rdim(M)) denote a coordinate chart with W  U. Then the pair ðEjW ; ðj; lU Þ: EjW ! RdimðMÞ  Rn ¼ RdimðMÞþn Þ is a coordinate chart for E. Remark 3.6 The definition given above endows each fiber of E with a canonical vector space structure. To see how this comes about, note first that each fiber is diffeomorphic to Rn; this being an obvious necessary condition. Second, the R-action on E gives an R-action on each fiber, and this gives R-action for the vector space structure. Third, the zero section oˆ gives each fiber an origin for the vector space structure. Granted these points, I need only describe the notion of vector addition on each fiber. This is done using a local trivialization. To elaborate, let p 2 M and let U  M containing p denote an open set with a local trivializing map jU: EjU ! U  Rn. Write the latter map as v ! (p(v), lU(v)). If v and v0 are 1 1 in Ejp, define v þ v0 to be lU,p (lU,p(v) þ lU,p(v0 )) where lU,p is the inverse of the n diffeomorphism from Ejp to R that is defined by lU. To see that this makes sense, I must demonstrate that the addition so defined does not depend on the chosen set U. The point here is that a given point p may lie in two sets, U and U0 , that come with corresponding maps lU and lU0 . I need

3.2

The standard definition

to prove that the same point v þ v0 is obtained when using lU0 . Now, this follows if and only if 1 1 0 0 lU 0;p ðl1 U;p ðe þ e ÞÞ ¼ lU 0;p ðlU;p ðeÞÞ þ lU 0;p ðlU;p ðe ÞÞ

for all pairs e and e0 2 Rn. This is to say that the map ¨

cU0; U ¼ lU0 ðl1 U Þ: ðU

U 0 Þ  Rn ! Rn

must be linear in the Rn factor; it must have the form (p, v) ! gU0 ,U(p)v where gU0 ,U is a smooth map from U U0 to Gl(n; R). That such is the case is a consequence of the next lemma. ¨

Lemma 3.7 A map c: Rn ! Rn with the property that c(rv) ¼ rc(v) for all r 2 R is linear. @ Proof of Lemma 3.7 Use the Chain rule to write @t cðtvÞ ¼ c* jtv ðvÞ. This is also equal to c(v) since c(tv) ¼ tc(v). Thus, c*jtv(v) ¼ c(v). Since the right-hand side of this last equality is independent of t, so is the left-hand side. Thus, the map t ! c*jtv (v) is independent of t, and so equal to c*j0(v). As a consequence, c(v) ¼ c*j0(v). The map gU0 ,U: U U0 ! Gl(n; R) is said to be a bundle transition function for the vector bundle.

¨

3.2

The standard definition

The observation that the map cU0 ,U depicted above is linear in the Rn factor suggests an alternate definition of a vector bundle. This one, given momentarily, is often the more useful of the two; and it is the one you will find in most other books. In this definition, a real vector bundle of fiber dimension n is given by the following data: First, a locally finite open cover, U, of M. Second, an assignment of a map, gU0 ,U: U U0 ! Gl(n; R) to each pair U, U0 2 U subject to two 1 constraints. First, gU,U0 ¼ gU0 ,U . Second, if U, U0 and U00 are any three sets 0 00 from U with U U U 6¼ , then gU0 ,U gU,U0 0 gU00 ,U0 ¼ i. By way of notation, the map gU0 ,U is deemed the bundle transition function. Meanwhile, the constraint gU0 ,U gU,U0 0 gU00 ,U ¼ i is said to be a cocycle constraint. Granted this data, the bundle E is defined as the quotient of the disjoint union n n 0 0 0 U2U (U  R ) by the equivalence relation that puts (p , v ) 2 U  R equivalent n 0 0 to (p, v) 2 U  R if and only if p ¼ p and v ¼ gU0 ,U(p)v. To connect this definition with the previous one, define the map p to send the equivalence class of any given (p, v) to p. The multiplication map is defined by the rule that has r 2 R sending the equivalence class of the point (p, v) to (p, rv). Finally, any given U 2 U has the corresponding map lU that is defined as

27

¨

¨

¨

\

3 : Introduction to vector bundles

follows: If p 2 U, then any given equivalence class in p1(p) has a unique representative (p, v) 2 U  Rn. This equivalence class is sent by lU to v 2 Rn. The smooth structure on a vector bundle E defined this way is that associated to the following sort of coordinate atlas: If necessary, subdivide each set U 2 U so that the resulting open cover consists of a set U0 where each set U 2 U0 comes with a smooth coordinate map jU : U ! Rdim(M) that embeds U as the interior of a ball. This is to say that the collection {(U, jU : U ! Rdim(M))}U2U0 consists of a coordinate atlas for M. This understood, then the collection {(EjU, (jU, lU): EjU ! Rdim(M)  Rn)} defines a coordinate atlas for E. Note that the coordinate transition function for the intersection of two such charts, thus for EjU 0 U, sends a pair (x, v) 2 jU(U U 0 )  Rn to the point with Rdim(M) coordinate (jU 0  jU1)(x) 2 jU0 (U U 0 ), and Rn coordinate gU 0 ,U(x)v. Thus, the bundle transition functions {gU0 ,U}U,U0 2U0 help to determine the smooth structure on E. The data consisting of the cover U with the bundle transition functions {gU0 ,U}U,U0 2U is called cocycle data for the bundle E. ¨

¨

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28

3.3

The first examples of vector bundles

What follows are some first examples of vector bundles.

3.3.1

The trivial bundle

The vector bundle M  Rn over M with fiber Rn is called the trivial Rn bundle for what I hope is an obvious reason. It is also often called the product bundle.

3.3.2

The Mobius bundle

Introduce the standard angular coordinate y 2 R /(2pZ) for S1. With S1 viewed as the unit radius circle about the origin in R2, a given angle y maps to the point (cos(y), sin(y)). Let E  S1  R2 denote the subset of points (y, (v1, v2)) such that cos(y)v1 þ sin(y)v2 ¼ v1 and sin(y)v1  cos(y)v2 ¼ v2. This defines a bundle over S1 with fiber dimension 1.

3.3.3

The tautological bundle over RPn

Define RPn as Sn/{1}. A bundle E ! RPn with fiber R is defined to be the quotient (Sn  R)/{1}. Alternately, E is the set (x, g) such that x 2 RPn and g is the line through the origin with tangent vector proportional to x.

3.4

3.3.4

The tangent bundle

Bundles over the Grassmannians

Fix positive integer m> n and introduce from Chapter 1.6 the manifold Gr(m; n) whose points are the n-dimensional subspaces of Rm. Over Gr(m; n) sits an essentially tautological Rn-bundle that is defined to be the set of pairs {(V, v) : V  Rm is an n-dimensional subspace and v 2 V}. Let E denote this bundle. The projection map p: E ! Gr(m; n) sends a given pair (V, v) to the n-dimensional subspace V 2 Gr(m; n). The R-action on E has r 2 R sending the pair (V, v) to the pair (V, rv). The map oˆ sends V 2 Gr(m; n) to the pair (V, 0) where 0 2 Rm is the origin. Let O V  Gr(m; n) denote the subspace of n-dimensional subspaces V0 which are mapped isomorphically via the orthogonal projection V : Rm ! V. The set U ¼ O V has the required map lU : p1(U) ! Rn, this defined as follows: Fix a basis for V so as to identify V with Rn. Granted this identification, define the map lO V so as to send any given pair (V0, v) 2 EjO V to V(v0 ) 2 V ¼ Rn. I leave it as an exercise to verify that E as just defined has a suitable smooth structure. This bundle E is called the tautological bundle over Gr(m; n).

3.3.5

Bundles and maps from spheres

The next example uses cocycle data to define a vector bundle. To start, let M denote the manifold in question and let m denote its dimension. Fix a finite set   M. Assign to each p 2  the following data: First, a smooth map gp: Sm1 ! Gl(n; R). Second, a coordinate chart (U, jp: U ! Rm) with p 2 U such that jp(p) ¼ 0 2 Rm. Fix r > 0 and small enough so that the open ball, B, of radius r about the origin in Rm is in the image of each p 2  version of jp; and so that the collection {Up ¼ j1 p (B)}p2 consists of pairwise disjoint sets. To construct a vector bundle from this data, use the cover given by the open sets U0 ¼ M and {Up}p2. The only nonempty intersections for this cover are those of the form U0 Up ¼ j1 p (B{0}). Define the bundle transition function g0,p: U0 Up ! Gl(n; R) on such a set to be the map x ! gp(jjp(x)j1jp(x)). As there are no nonempty intersections between three distinct sets in the cover, there are no cocycle conditions to satisfy. As a consequence, this data is sufficient to define a vector bundle over M having fiber dimension n. ¨

¨

3.4

The tangent bundle

Every smooth manifold comes a priori with certain canonical vector bundles, one of which is the tangent bundle, this a bundle whose fiber dimension is that

29

3 : Introduction to vector bundles

of the manifold in question. The tangent bundle of a manifold M is denoted by TM. What follows is a cocycle definition of TM. Let n denote the dimension of M. Fix a locally finite coordinate atlas, U, for M. The collection of open sets from the pairs in U is used to define the tangent bundle by specifying the bundle transition functions {gU0 ,U: U0 U ! Gl(n; R)}(U,j),(U0 ,j0 )2U . Fix a pair (U, jU) and (U0 , jU0 ) from the atlas U. To define gU0 ,U, introduce the coordinate transition function cU0 ,U ¼ jU0  jU1, a diffeomorphism from jU(U0 \ U)  Rn to jU0 (U0 \ U)  Rn. The differential of this map, cU0 ,U*, is a map from cU(U0 \ U) to Gl(n; R). This understood, set gU0 ,U(p) ¼ cU0 ,U*jjU(p) for p 2 U0 \ U. The cocycle condition gU0 ,UgU,U0 0 gU0 0 ,U0 ¼ i is guaranteed by the Chain rule by virtue of the fact that cU0 ,UcU,U0 0 cU0 0 ,U0 is the identity map from cU0 (U0 \ U) to itself. This definition appears to depend on the choice of the atlas U from the equivalence class that defines the smooth structure. As explained next, the same bundle TM appears for any choice of atlas from this equivalence class. To see why this is, agree for the moment to use TU to denote the version of TM defined as above using the atlas U. Suppose that V is a coordinate atlas that is equivalent to U. Let TV denote the version of M’s tangent bundle that is defined as above from V. I now define a coordinate atlas W as follows: The collection W is the union of two coordinate atlases. The first, W U consists of all pairs of the form (U V, jU) such that (U, jU) is from U and (V, jV) is from V. The second atlas, W V , consists of all pairs of the form (U V, jV) with U, V and jV as before. Thus, if (U, jU) 2 U and (V, jV) 2 V, then the open set U V appears in two pairs from W, one with the map jU and the other with the map jV. Use TW U , TW V and TW to denote the versions of the tangent bundle that are defined respectively by W U , W V and W. It follows directly from the definition of the equivalence relation that TW ¼ TW U ¼ TW V ; indeed, such is the case because any given point in M is contained in some version of U V. It also follows directly from the equivalence relation that TW U ¼ TU and TW V ¼ TV. What follows is an important point to make with regards to the definition of TM: If U  M is any given coordinate chart jU : U ! Rn, then there is a canonically associated bundle trivialization map from TMjU to U  Rn. The latter is denoted by jU*. ¨

¨

¨

¨

¨

30

3.4.1

A geometric interpretation of TM

Suppose that m  n and that s: M ! Rm is an immersion. Recall that this means the following: Let (U, j: U ! Rn) denote any given coordinate chart on M. Then s 8 (j1): Rn ! Rm has injective differential. The map s gives the tangent bundle to M a geometric interpretation whose description follows. To start, fix a locally finite coordinate atlas U for M. Given p 2 M, fix a chart

3.5

Tangent bundle examples

(U, j) 2 U with p 2 U; and let p  Rm denote the n-dimensional linear subspace given by the differential of s  (j1). The subspace p does not depend on the chosen chart containing p, nor on the atlas U representing the smooth structure. This understood, it follows that p is canonically isomorphic as a vector space to TMjp. As a consequence, TM can be viewed as sitting in M  RN as the set of pairs {(p, v) 2 M  RN : v 2 p}. This view of TM identifies the projection p: TM ! M with a restriction of the tautological projection from M  Rm to M. Meanwhile, the R-action on TM and the vector space structure on each fiber is that inherited from the R-action on Rm and the vector space structure of Rm. Suppose that m  n and that c: Rm ! Rmn is a smooth map. Suppose, in addition, that a 2 Rmn is a regular value of c. Recall from Chapter 1 that M ¼ c1(a) is a smooth, n-dimensional submanifold of Rm. If p 2 M, then the linear subspace p is the kernel of the c*jp. Thus, the tangent bundle TM is the subset {(p, v) : p 2 M and c*jp(v) ¼ 0}. As a parenthetical remark, note that the map from Rm  Rm to Rmn  Rmn that sends any given pair (p, v) to C(p, v) ¼ (c(p), c*jpv) has (a, 0) 2 Rmn  Rmn as a regular value. As noted in the previous paragraph, C1(a, 0) is TM. Thus, TM sits in Rm  Rm as a submanifold. It is left as an exercise to check that its smooth structure as M’s tangent bundle is the same as that coming from the identification just given as the inverse image of a regular value of C.

3.5

Tangent bundle examples

What follows are some tangent bundle examples.

3.5.1

The manifold Rn

The manifold Rn has a canonical coordinate atlas, this the atlas with one chart, and with the associated map to Rn being the identity. Using this coordinate atlas U ¼ {(Rn, the identity map)} as a fiducial atlas identifies TRn with the product bundle Rn  Rn with the bundle projection p given by projection to the left Rn factor.

3.5.2

An open subset of Rn

An open set M  Rn has the fiducial coordinate atlas U ¼ {(M Rn, the identity map M ! Rn)}. Using this chart identifies M’s tangent bundle with M  Rn

31

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3 : Introduction to vector bundles

also. For example, the group Gl(n; R) sits as an open set in the Euclidean space M(n; R). Thus, TGl(n; R) can be canonically identified with Gl(n; R)  M(n; R). Likewise, TGl(n; C) can be canonically identified with Gl(n; C)  M(n; C).

3.5.3

The sphere

The sphere Sn  Rnþ1 is the set of vectors x 2 Rnþ1 with jxj ¼ 1. This is c1(1) where c: Rnþ1 ! R is the map sending x to jxj2. The differential of this map at a given point x 2 Rnþ1 sends v 2 Rnþ1 to x·v where · here denotes the Euclidean inner product. This being the case, TSn sits in Rnþ1  Rnþ1 as the set of pairs {(x, v) : jxj ¼ 1 and v is orthogonal to x}.

3.5.4

A surface in R3

Take g  0 and construct a surface of genus g embedded in R3 as described in Chapter 1.3 using a suitable function h: R2 ! R. Recall that this surface is the level set with value 0 for the function (x, y, z) ! z2  h(x, y). The tangent bundle to this surface sits in R3  R3 as the set of pairs (p ¼ (x, y, z), v ¼ (vx, vy, vz)) with @h  @h  z2  hðx; yÞ ¼ 0 and also 2zvz  vx  vy ¼ 0:   @x ðx;yÞ @y ðx;yÞ

3.5.5

The group Sl(n; R)

This manifold sits inside M(n; R) as c1(1) where c is the map c from M(n; R) to R given by c(m) ¼ det(m). The differential of this map at an invertible matrix m sends a 2 M(n; R) to det(m)tr(m1a). The kernel of this is the vector subspace in M(n; R) consisting of matrices {a ¼ mc : tr(c) ¼ 0}. Thus, the tangent bundle to Sl(n; R) sits in M(n; R)  M(n; R) as the submanifold consisting of the pairs {(m, mc) : det(m) ¼ 1 and tr(c) ¼ 0}.

3.5.6

The group O(n)

The group O(n) sits in M(n; R) as c1(i) where c is the map from M(n; R) to the Euclidean space Sym(n; R) of n  n symmetric matrices that sends a matrix m to mTm. The differential of c at m sends a 2 M(n; R) to mTa þ aTm. The kernel of this map at m 2 SO(n) is the vector subspace {a ¼ mc : cT ¼ c}. Thus, TO(n) sits in M(n; R)  M(n; R) as the submanifold of pairs {(m, mc) : mTm ¼ 1 and cT ¼ c}.

3.6

3.5.7

The cotangent bundle

The groups U(n) and SU(n)

The group U(n) sits in M(n; C) as the inverse image of i via the map to the space Herm(n) of n  n Hermitian matrices that sends a matrix m to m{m. The kernel of the differential of this map at m 2 U(n) consists of the matrices a 2 M(n; C) that can be written as a ¼ mc where c{ ¼ c. This understood, TU(n) sits in M(n; C)  M(n; C) as the submanifold of pairs {(m, mc): m{m ¼ i and c{ ¼ c}. Meanwhile, TSU(n) sits in M(n; C)  M(n; C) as the submanifold of pairs {(m, mc) : both m{m ¼ i and det(m) ¼ 1, and both c{ ¼ c and tr(c) ¼ 0}.

3.6

The cotangent bundle

The cotangent bundle is the second of the two canonical bundles that are associated to M. This one is denoted T*M; and it has fiber dimension n ¼ dim(M). What follows is the cocycle definition of T*M. Fix a coordinate atlas U for M. As with the tangent bundle, the collection of open sets that appear in the pairs from U supply the open cover for the cocycle definition. To define the bundle transition functions, suppose that (U, jU) and (U0, jU0 ) are two charts from U. Reintroduce the coordinate transition function cU0 ,U ¼ jU0  jU1. As noted a moment ago, this is a diffeomorphism from one open set in Rn to another. In particular its differential cU0 ,U* maps cU(U0 U) to Gl(n; R). This understood, the bundle transition function gU0 ,U for the cotangent bundle is defined so that its value at any given point p 2 U0 U is the transpose of the inverse of the matrix cU0 ,U*jjU(p). Said differently, any given bundle transition function for T*M is obtained from the corresponding bundle transition function for TM by taking the transpose of the inverse of the latter. The same argument that proves TM to be independent of the representative coordinate atlas proves that T*M is likewise independent of the representative atlas. Here is a final remark: Let U  M denote any given coordinate chart, and let jU denote the corresponding map from U to Rn. It follows from the definition of T*M that the latter has a canonically associated bundle trivialization map from T*MjU to U  Rn. The inverse of this trivialization map, a map from U  Rn to T*MjU, is denoted by jU*. The only example of the cotangent bundle I’ll give now is that of Rn, in which case the canonical coordinate atlas U ¼ {(Rn, the identity map)} identifes TRn with the product bundle Rn  Rn. More cotangent bundles examples are given in later chapters.

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3 : Introduction to vector bundles

3.7

Bundle homomorphisms

Let p: E ! M and p0: E0 ! M denote a pair of vector bundles. A homomorphism h: E ! E0 is a smooth map with the property that if p 2 M is any given point, then h restricts to Ejp as a linear map to E0 jp. An endomorphism of a given bundle E is a homomorphism from E to itself. An isomorphism between given bundles E and E0 is a bundle homomorphism that maps each p 2 M version of Ejp isomorphically to the vector space E0 jp. For example, the definition of a vector bundle given in Chapter 1.1 implies the following: If E is a vector bundle with fiber dimension n and point p 2 M a given point, then p is contained in an open set U such that there is a bundle isomorphism jU : EjU ! U  Rn. These local trivializations can be used to give a somewhat different perspective on the notion of a bundle homomorphism. To set the background, suppose that E and E0 are two vector bundles over M of fiber dimensions n and n0, respectively. Let p 2 M a given point. There is then an open neighborhood U  M of p with respective bundle isomorphisms jU : EjU ! U  Rn and 0 j0U : E0 jU ! U  Rn . A given bundle homomorphism h: E ! E0 is such that the 0 composition j0U 8 h 8 jU1 is a map from U  Rn to U0  Rn that can be written as (x, v) ! (x, hU(x)v) where x ! hU(x) is a smooth map from the set U to the 0 0 vector space, Hom(Rn; Rn ), of linear maps from Rn to Rn . Note that if U0 is another open set in M with U0 U 6¼  and with corresponding bundle isomorphisms jU0 for EjU0 and j0U0 for E0 jU0 , then there is a corresponding hU0 . The latter is determined by hU on U0 U as follows: Introduce the respective bundle transition functions, gU0 ,U: U0 U ! Gl(n; R) for E and g0U0 ,U: U0 U ! Gl(n0 ; R) for U0 . Then hU0 ¼ gU0 ,U hU(g0U0 ,U)1 at the points in U0 U. ¨

¨ ¨

¨

¨

Example 3.8 This second picture of a vector bundle isomorphism can be used to tie up a loose end that concerns Chapter 3.2’s cocycle definition of a vector bundle. This loose end concerns the notion of the refinement of an open cover. To set the stage, let U denote a given open cover of M. A second open cover, U0 , is said to be a refinement of U when each set from U0 is contained entirely in at least one set from U. This notion of refinement is often used when comparing constructions that are defined using distinct open covers. The point being that if U and V are any two open covers, then the collection {U V}U2U, V2V is a simultaneous refinement of both. In any event, suppose that n  0 is a given integer, that U is a finite open cover, and that {gU,U0 : U U0 ! Gl(n; R)}U,U0 2U supply the cocycle data needed to define a vector bundle over M with fiber Rn. Let E ! M denote the resulting vector bundle. Now suppose that V is a second finite open cover that refines U. ¨

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34

3.8

Sections of vector bundles

Fix a map u: V ! U such that V  u(V) for all V 2 V. This done, then the data {gU(V),U(V0 ): V V0 ! Gl(n; R)}V,V0 2V supplies the cocycle data for a vector 0 bundle, E ! M. As it turns out, there is a canonical isomorphism from E0 to E. To see this isomorphism, recall that the cocycle definition of E0 defines the latter as the quotient of V2V (V  Rn) by an equivalence relation. Likewise, E is the quotient of U2U (U  Rn). Let V 2 V. The isomorphism in question takes the equivalence class defined by (p, v) 2 V  Rn to that defined by the pair (p, v) with the latter viewed as a pair in u(V)  Rn. It is an exercise to unwind the definitions to see that this map is, as claimed, an isomorphism. This canonical isomorphism is used implicitly to identify the bundle E with the bundle E0 that is defined in this way using a refinement of the cover U. ¨

Example 3.9 What follows describes a Lie group application of the notion of vector bundle isomorphism. To start, suppose that G is one of the Lie groups that were introduced in Chapter 2.3 or Chapter 2.5. In each case, the tangent bundle TG is isomorphic to the product bundle G  Rdim(G). In the case G ¼ Gl(n; R) or Gl(n; C), this comes about because G is an open set in the Euclidean space M(n; R) or M(n; C) as the case may be. The tangent bundle to Sl(n; R) was identified in Chapter 3.5 with the submanifold in Sl(n; R) M(n; R) of pairs of the form (m, mc) where c has trace equal to 0. Let M0(n; R) denote the vector space of such matrices. This is a Euclidean space, and the map from Sl(n; R)  M0(n; R) to TSl(n; R) that sends a pair (m, c) to (m, mc) is a bundle isomorphism. The tangent space to SO(n) was identified with the set of pairs (m, mc) 2 SO(n)  M(n; R) such that cT ¼ c. Let A(n; R) denote the vector space of such n  n, skew-symmetric matrices. The map that sends a pair (m, c) in the product vector bundle SO(n)  A(n; R) to (m, mc) 2 TSO(n) is also a bundle isomorphism. There are analogous isomorphisms for the groups U(n) and SU(n). To describe the former case, let A(n; C) denote the vector space of anti-Hermitian matrices. Thus, c 2 A(n; C) if and only if c{ ¼ c. The map that sends (m, c) 2 U(n) A(n; C) to the pair (m, mc) 2 TU(n) is a bundle isomorphism. To consider the case of SU(n), introduce A0(n; C) to denote the subvector space in A(n; C) of trace zero matrices. The same map restricts to the subspace SU(n)  A0(n; C) to give a bundle isomorphism from the latter space to TSU(n).

3.8

Sections of vector bundles

Let p: E ! M denote a given vector bundle. A section of E is a smooth map s: M ! E such that p 8 s is the identity map on M. Thus a section assigns to each point p 2 M a point in Ejp. For example, the zero section oˆ that appears in the

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definition given in Chapter 3.1 is an example. Note that the space of sections is a linear space. Moreover, it is linear over C1(M; R) in the following sense: If s is any given section and f : M ! R any given function, then f s is also a section; this is the section that assigns f (p)s(p) 2 Ejp to p 2 M. The space of sections of E is denoted by C1(M; E) in what follows. To see what a section looks like, suppose that U  M is an open set with an isomorphism jU : EjU ! U  Rn. Here, n is the fiber dimension of E. Let s denote any given section of E. Then jU 8 s: U ! U  Rn has the form x ! (x, sU(x)) where sU is a smooth map from U to Rn. Note that if U0  M is an open set with a trivialization jU0 from EjU0 ! U0  Rn, then there is a corresponding map sU0 , with the latter determined by sU over U0 U by the rule sU0 ¼ gU0 ,UsU where gU0 ,U: U0 U ! Gl(n; R) again denotes the bundle transition function. This identification can be used to see why the vector space of sections of any given bundle E is infinite dimensional. Indeed, fix U  M where there is a bundle isomorphism jU : EjU ! U  Rn. Let sU : U ! Rn denote any given map which is zero on the complement of a compact set in U. A section, s, of E can be defined now as follows: Set s ¼ oˆ on EjMU and set s on U to be the section given by p ! j1 U ((p, sU(p)). By the way, a comparison of the definition of a section with that of a bundle homomorphism finds that a section of E gives a bundle homomorphism from the trivial bundle M  R to E, and vice versa. The correspondence identifies any given section s with the homomorphism that send (p, r) 2 M  R to rs(p). Meanwhile, a homomorphism h: M  R ! E gives the section p ! hjp(1) where 1 2 R. If U  M is any open set, a set {sk}1kn of sections of EjU are said to define a basis of sections of EjU if the set {skjp}1kn 2 Ejp is linearly independent for each p 2 U where E has fiber dimension n. Thus they define a basis of Ejp. A basis of sections gives an isomorphism from the product bundle U Rn to EjU, this the homomorphism that sends a given (p, (r1, . . . , rn)) 2 U Rn to the point P 1kn rk sk(p). Conversely, a basis of sections for EjU defines an isomorphism from EjU to U  Rn, this the homomorphism that sends any given point v 2 Ejp to P (p, (r1(v), . . . , rn(v)) 2 U Rn where v ¼ 1kn rk sk(p). A bundle E ! M is isomorphic to the product bundle if and only if there is a basis of sections for the whole of M. ¨

¨

36

3.9

Sections of TM and T*M

A section of the tangent bundle of M is called a vector field. The space of sections of TM, thus C1(M; TM), can be viewed as the vector space of derivations on the algebra of functions on M. To elaborate, remark that the space C1(M; R) of smooth functions on M is an algebra with addition given by

3.9

Sections of TM and T*M

(f þ g)(p) ¼ f(p) þ g(p) and multiplication given by (fg)(p) ¼ f(p)g(p). A derivation is by definition a map, L, from C1(M; R) to itself that obeys L(f þ g) ¼ Lf þ Lg and L(fg) ¼ Lf g þ f Lg. Let v denote a given section of TM. The action on a function f of the derivation defined by v is denoted by v(f). To see what this looks like, go to a coordinate chart U  M with coordinate map jU : U ! Rn with n here set equal to the dimension of M. Use jU : TMjU ! U  Rn to denote the associated isomorphism between TMjU * and the product bundle. Write jU  v as the section of the product bundle given by p ! (p, vU ¼ (v1, . . . , vn): U ! Rn). Meanwhile, write fU ¼ f 8 jU1. The P @ analogous (vf)U is given by ðvfÞU ¼ 1in vi ð@x f U Þ. The fact that this definition i is consistent across coordinate charts follows using the Chain rule and the bundle transition function for TM. The function vf is often called the Lie derivative of f along the vector field v. It is also called the directional derivative of f along v because the formula that P writes (vf)U as 1in vi ð@x@ i f U Þ identifies (vf)U with the directional derivative of the function fU on Rn along the vector field with components (v1, . . . , vn). The observation that vector fields give derivations motivates the notation P @ whereby vU is written in terms of its components as 1in vi @xi ; and it n o @ motivates the notation that uses @x as a basis of sections for TRn. The i 1in

map (j1 U )* takes this basis to a basis of sections of TM over U. To see how a derivation of C1(M; R) gives a vector field, let L denote a given derivation. Let U  M denote an open set where there is a coordinate map jU to Rn. For each k 2 {1, . . . , n}, let fk ¼ xk  jU. Define ck to be the function on U given by Lfk. It then follows using Taylor’s theorem with remainder and the P Chain rule that L acts on any given function, Lf, on U as Lf ¼ 1kn ckvk, where vk is the vector field on Unthat o corresponds via the trivialization n @ (jU) : TMjU ! U  R to the basis @xi . * 1in A section of T*M is called a 1-form. Any given function, f, on M defines a 1-form, this denoted by df. The latter is said to be the exterior derivative of f. There is an associated linear function on TM defined as follows: Let p 2 M and let v 2 TMjp. Let v denote any given vector field on M with vjp ¼ v. The linear function df send v to (vf)jp. To see what this looks like, let U  M denote a coordinate chart and let jU : U ! Rn denote the corresponding map. Write n f 8 j1 U as fU again, this a function on a domain in R . Meanwhile, jU defines a canonical trivialization of T*MjU.Then  the image  of the section df via @f U @f U ; . . . ; this trivialization is the section x ! x; @x @xn x of the product bundle 1 n j(U)  R . P @h This last observation motivates the writing of dh as 1in @x dxi when h is a i n given function on R . The notation indicates that the exterior derivatives of the coordinates functions {xi}1in define the basis {dxi}1in of sections for T*Rn. This basis corresponds via jU’s trivialization of T*MjU to a basis of sections of T*MjU.

37

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Additional reading  Introduction to Smooth Manifolds, John M. Lee, Springer, 2003.  Tensor Analysis on Manifolds, Richard L. Bishop and Samuel Goldberg, Dover Publications, 1980.  Characteristic Classes, John W. Milnor and James D. Stasheff, Princeton University Press, 1974.  Manifolds and Differential Geometry, Jeffrey M. Lee, American Mathematical Society, 2009.  Vector Bundles, Volume 1: Foundations and Stiefel–Whitney classes, Howard Osborn, Academic Press, 1983.  Differential and Riemannian Manifolds, Serge Lange, Springer, 1995.

4

Algebra of vector bundles

Any linear operations that can be done to generate a new vector space from some given set of initial vector spaces can be done fiber-wise with an analogous set of vector bundles to generate a new vector bundle. What follows describes the most important examples.

4.1

Subbundles

Let p: E ! M denote a vector bundle. A submanifold E0  E is said to be a subbundle of E if, for each p 2 M, the subspace E0 jp ¼ (E0 Ejp) is a vector subspace. An equivalent definition is as follows: The submanifold E0  E is a subbundle if it is the image via a bundle homomorphism that is injective on each fiber. To see that the structure of a subbundle E0  E near any given point p 2 M, fix an open set U  M that contains p and that comes with an isomorphism jU : EjU ! U  Rn. The image via jU of E0 jU is an n0 -dimensional subbundle of the product bundle U  Rn. Let V0  Rn denote the jU image of E0 jp, this an n0 -dimensional vector subspace of Rn. Fix a basis for V0 so as to identify it with 0 0 Rn . Let pV 0 : Rn ! V0 ¼ Rn denote the orthogonal projection. Then there is a neighborhood U0  U of p such that the composition pV0  jU maps E0 jU0 0 isomorphically to the trival bundle U0  V0 ¼ U0  Rn. By way of an example, suppose that M sits in RN as a submanifold. Then TM sits in the product bundle M  RN as a subbundle. To see a second example, fix positive integers m > n and introduce from Chapter 1.6 the Grassmannian Gr(m; n) whose points are the n-dimensional vector subspaces of Rm. Introduce from Chapter 3.3 the tautological Rn-bundle E ! Gr(m; n), this the set of pairs {(V, v) 2 Gr(m; n)  Rn : v 2 V}. This bundle is defined to be a subbundle of the product bundle Gr(m; n)  Rn. As it turns out, any given bundle p: E ! M can be viewed as a subbundle of some product bundle M  RN for N very large. What follows explains why this ¨

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4 : Algebra of vector bundles

is true in the case when M is compact. To start, fix a finite open cover U of M such that each set U  U has an associated bundle isomorphism jU : EjU ! U  Rn. By enlarging the set U if necessary, one can also assume that there is an associated, subordinate partition of unity—this is a set of functions {wU : U ! [0, 1)}U2U P such that any given wU is zero on MU and such that U2N wU ¼ 1 on M. Label the elements in U as {Ua}1aK. Now define a bundle homomorphism c: E ! M  RKn by the following rule: Write RKn as K factors of Rn. A given point v 2 E is sent to the point in M RKn whose K’th Rn factor is wUk(p(v))jUk(v). By construction, the map so defined is fiber-wise injective, and so c(E) sits in M  RKn as a subbundle.

4.2

Quotient bundles

Suppose that V is a vector space and V0  V a vector subspace. One can then define the quotient space V / V0, this the set of equivalence classes that are defined by the rule that has v u when v  u 2 V0. The quotient V/ V0 is a vector space in its own right, its dimension is dim(V)  dim(V0 ). Note that in the case when V is identified with Rn, the orthogonal projection from Rn to the orthogonal complement of V0 identifies the quotient bundle V/ V0 with this same orthogonal complement. There is the analogous quotient construction for vector bundles over a given manifold. To elaborate, suppose that p: E ! M is a vector bundle with fiber dimension n and E0  E is a subbundle with fiber dimension n0  n. The quotient bundle E /E0 is defined so that its fiber over any given point p 2 M is Ejp / E0 jp. The projection from E /E0 is induced by that of E to M, and the R-action is induced from the corresponding action on E. To see the structure of E/E0 near a given point p 2 M, suppose that U  M is an open set that contains p and that comes with a bundle isomorphism jU : EjU ! U  Rn. The latter restricts to E0 jU so as to map E0 jU isomorphically onto some n0 -dimensional subbundle of the product bundle U  Rn. Set jU(E0 jp) ¼ V0, this an n0 -dimensional vector subspace of Rn. Let pV0 as before denote the orthogonal projection from Rn to V0. As noted in Chapter 4.1, there is a neighborhood U0  U of p such that j0U0 ¼ pV0  jU maps E0 jU0 isomorphically to the trivial bundle U  V0. Let V0 ⊥  Rn denote the orthogonal comple⊥ ment of V0, and let pV0 denote the orthogonal projection from Rn to V0 ⊥. 1 0 and let Introduce p: U  V0 ! V0 ⊥ to denote the composition p⊥ V0  (j U0 ) ⊥ ⊥ 00 0 0 0 0 jU ¼ pV0  jU  p  j U0 , this a map from EjU0 to U  V . By construction, this map sends vectors in E0 jU0 to zero. Meanwhile, it is surjective at p, and this implies that it is surjective on some neighborhood U00  U0 of p. This

4.3

The dual bundle

understood, it induces an isomorphism (E/E0 )jU0 ! U0  V0 ⊥. Choose a basis for 0 V0 ⊥ to identify the latter with Rnn and the resulting map from (E/E0 )jU0 to U0  0 Rnn gives the requisite local isomorphism. By way of an example, suppose that M is realized as a submanifold in Rm for some m > n ¼ dim(M). As explained in Chapter 3.4, the tangent bundle TM appears as a subbundle of M  Rn. The quotient bundle (M  Rn)/ TM is said to be the normal bundle of M in Rn. Consider the case when M  R3 is a surface of genus g as constructed in Chapter 1.3 using a function h: R2 ! R. Recall that M in this case is the locus of points where z2  h(x, y) ¼ 0. The normal bundle in this case has fiber dimension 1, and orthogonal projection in R3 identifies it with the subbundle in M  R3 given by 9 80 0 11     = < @ @h  @ AA @ðx; y; zÞ; v ¼ r@z @  @h   :r2R :  ; : @z @x  @x @y  @y ðx;yÞ

4.3

ðx;yÞ

The dual bundle

Let V denote a vector space of dimension n. The dual vector space, V*, is the vector space of linear maps from V to R. There is a corresponding notion of the dual of a vector bundle. To say more, let p: E ! M denote the vector bundle in question. Its dual is denoted by E*; it is defined so that the fiber of E* over any given point p 2 M is the dual of the vector space Ejp. To elaborate, E* is, first of all, the set of pairs {(p, l) : p 2 M and l: Ejp ! R) is a linear map}. The bundle projection p: E* ! M sends any given pair (p, l) 2 E* to the point p. The R action is such that r 2 R acts to send (p, l) 2 E* to (p, r l). The definition also requires a suitable map to Rn near any given point; a map that restricts to each fiber as an R-equivariant diffeomorphism. For this purpose, let U  M denote an open set where there is a smooth map lU : EjU ! Rn that restricts to each fiber as an R-equivarant diffeomorphism. The corresponding map l* U for E*jU is best viewed as a map to the vector space of linear functions on Rn. (This, of course, is a copy of Rn.) Granted this view, the map l*U is defined by the following requirement: Let p 2 U and let e 2 Ejp. Then n l* U maps (p, l) to the linear functional on R whose value on lU(e) is the same as that of l on e. 0 Note that this definition of l* U has the following consequence: Let U  M 0 denote an open set with U U 6¼  and with an analogous map lU0 : EjU0 ! Rn. Introduce gU0 ,U : U0 U ! Gl(n; R) to denote the corresponding bundle n transition function. Given lU0 , there is a corresponding l* U0 : E*jU0 ! R , and

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4 : Algebra of vector bundles

a corresponding bundle transition function for E* that maps U0 U to Gl(n; R). The latter is the map (gU0 ,U1)T, the transpose of the inverse of gU0 ,U. The description of E* is not complete without saying something about its smooth structure. The latter comes about as follows: Let U ¼ {(U, j: U ! Rdim(M))} denote a coordinate atlas for M. No generality is lost by requiring that each set U from this atlas come with a map lU : EjU ! Rn that restricts to each fiber as an R-equivariant diffeomorphism. As noted earlier, the set EjU with the map (j, lU) gives a coordinate chart for E. This understood, the smooth structure for E* is that defined by the atlas given by {(E*jU, (j, l*U))}(U,j)2U . The coordinate transition functions for this atlas are smooth. Indeed, if (U0, j0 ) 2 U and U0 U 6¼ , then the coordinate transition function for the intersection of the charts E*jU E*jU0 ¼ E*jU U0 is the map ¨

¨

¨

¨

ðj0  ðjÞ1 ; ðgU0 ;U1 ÞT ð ÞÞ where gU0 ,U: U0 U ! Gl(n; R) is the bundle transition function for E over U U 0 . This is a smooth map, and so the map to Gl(n; R) given by the transpose of its inverse is also a smooth map. Note that if V is a vector space, then (V*)* has a canonical identification with V. Indeed, a vector v 2 V gives the linear function l ! l(v) on V*. By the same token, if E is a vector bundle, then (E*)* ¼ E. Perhaps the canonical example is the case when E is the tangent bundle TM; in this case, E* is the cotangent bundle, T*M. This can be seen by comparing the bundle transition functions. ¨

4.4

¨

42

Bundles of homomorphisms

The notion of the dual of a vector space, Hom(V; R), has a generalization as follows: Let V and V0 denote two vector spaces. Let Hom(V; V0 ) denote the set of linear maps from V to V0. This also has the structure of a vector space, one with dimension equal to the product of the dimensions of V and of V0. This can be seen by choosing a basis for V to identify it with Rdim(V) and likewise for V0 0 to identify it with Rdim(V ). A linear map from V to V0 appears now as a matrix with dim(V) columns and dim(V0 ) rows. The construction Hom( ; ) can be applied fiber-wise to a pair of vector bundles, E and E0 , over M, to give a new vector bundle, this denoted by Hom(E; E0 ). The fiber of Hom(E; E0 ) over any given point p 2 M is the vector space Hom(Ejp; E0 jp). If E has fiber dimension n and E0 fiber dimension n0 , then the fiber dimension of Hom(E; E0 ) is nn0 . To continue with the definition of the vector bundle structure, note that R-action is such that r 2 R acts on Hom(E; E0 )

4.6

The direct sum

so as to send a given point (p, l: Ejp ! E0 jp) to (p, rl). The required neighbor0 hood maps to Rnn are obtained as follows: Let p 2 M denote some given point and U  M a neighborhood of p with R-equivariant maps lU : EjU ! Rn and 0 0 l0U : E0 ! Rn . The requisite map lHom,U : Hom(E; E0 )jU ! Rnn is defined so as 1 to send any given point (p0 , l) to the point matrix l0 U  l  l1 U where here lU is n viewed as a linear map from R to Ejp0 . As for the smooth structure, take U as above so that it comes with a coordinate map j: U ! Rdim(M). This understood, then Hom(E; E0 )jU is a coordinate chart for Hom(E; E0 ) with the map that sends any given pair (x, l) to 0 (j(p), l0 U  l  lU1) 2 Rdim(M)  Rnn . To see that the coordinate transition 0 functions are smooth, it proves useful to view any given point m 2 Rnn as a 0 matrix m 2 Hom(Rn; Rn ). This understood, suppose that U0  M is another coordinate chart with map j0 : U0 ! Rdim(M) and with corresponding bundle 0 maps lU0 : EU0 ! Rn and lU0 0 : E0 jU0 ! Rn . Then the transition function sends a 0 given point (x, m) 2 j(U U0 )  Hom(Rn; Rn ) to the point with coordinates 0 1 0 0 0 ((j0  j1)(x), gU U0 )  Hom(Rn; Rn ). ,U(x) m gU0 ,U(x) ) 2 j (U Note that a section of Hom(E; E0 ) over M is nothing more than a bundle homomorphism from E to E0 . ¨

¨

4.5

Tensor product bundles

Fix vector spaces V and V0 with respective dimensions n and n0. The vector space N Hom(V*; V0 ) of linear maps from the dual space V* to V0 is denoted by V0 V. What follows gives a pedestrian way to view this vector space. Fix a basis {va}1an for V and a basis {va0 }1an0 for V0. A basis for the nn0 -dimensional N N vector space V0 V can be written as {va0 vb}1an0 ,1bn where the element N 0 va vb represents the linear map from V* to V0 that assigns to any given linear function l 2 V* the element l(vb) va0 2 V0. Note that a decomposable element in N V is given by a pair v0 2 V0 and v 2 V; it assigns to any given l 2 V* the V0 N element l(v)v0 2 V0. A decomposable element of this sort is written as v0 v. The N vector space V0 V is spanned by its decomposable elements. By analogy, if E0 ! M and E ! M Nare any two vector bundles, the corresponding tensor product bundle is E0 E ¼ Hom(E*; E0 ).

4.6

The direct sum

Let V and V0 denote a pair of vector spaces with respective dimensions n and n0 . L The direct sum V V0 is the vectors space of pairs (v, v0 ) 2 V  V0 where

43

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addition is defined by (v, v0 ) þ (u, u0 ) ¼ (u þ v, u0 þ v0 ) and the R-action has r 2 R sending (v, v0 ) to (rv, rv0 ). To see the bundle analog, let p: E ! M and p0 : E0 ! M denote a pair of vector L bundles. Their direct sum, E E0 , is also a bundle over M. It can be viewed as the subset in E  E0 of pairs {(v, v0 ) : p(v) ¼ p0 (v0 )}. As explained momentarily, L this subset is a submanifold and thus E E0 is a smooth manifold. The bundle L E0 to M sends (v, v0 ) to p(v), this by definition being the projection from E 0 0 same point as p (v ). The R-action has r 2 R acting on (v, v0 ) as (rv, rv0 ). To complete the definition, I need to exhibit a local trivialization on some neighborhood of any given point. To do this, let U  M denote a given open set with a 0 pair of maps lU : EjU ! Rn and lU0 : E0 jU0 ! Rn that restrict to the respective fibers as an R-equivariant diffeomorphism. The corresponding map for L 0 (E E0 )jU sends the latter to Rn  Rn and is given by (lU, lU0 ). It is a consequence L of these definitions that the L fiber over any given point p 2 M of E E0 is the vector space direct sum Ejp E0 jp. L To tie up a loose end, I need to explain why E E0 sits in E  E0 as a submanifold. For this purpose, take U as above, and let W  U denote an open subset with a smooth coordinate embedding j: W ! Rdim(M). The pair (j, lU) gives a coordinate embedding from EjW ! Rdim(M)  Rn. Likewise, the 0 pair (j, lW ) gives a coordinate embedding from E0 jW ! Rdim(M)  Rn. This understood, then (EjW  EjW0 ) has the coordinate embedding to 0 (Rdim(M)  Rn)  (Rdim(M)  Rn ) given by ((j, lW), (j, l0W)). The composition of this map with the map to Rdim(n) given by ((x, v), (x0 , v0 )) ! x  x0 has 0 as a regular value. This, the inverse image of 0, is a smooth submanifold. The L E0 )jW. AsLa parenthetical remark, it follows as a latter is, by definition (E 0 consequence that the map from (E E0 )jW to Rdim(M)  (Rn  Rn ) given by (j, (lU, lU0 )) is a coordinate embedding.

4.7

Tensor powers

Let V denote a given vector space. Recall that V* is the vector space of linear N maps from V to R. Fix k  1. As it turns out, the vector space V* V* is canonically isomorphic to the vector space of bilinear maps from V  V to R. This is to say the vector space of maps (v1, v2) ! f (v1, v2) 2 R with the property that f ( , v2) is linear for any fixed v2 2 V and f (v1, ) is linear for any fixed v1 2 V. The identification is such as to send any given decomposable element N N l1 l2 2 V* V* to the function (v1, v2) ! l1(v1)l2(vN 2). N N By the same token, if k  1 is any given integer, then k V* ¼ V* . . . V* is canonically isomorphic as a vector space to the vector space of k-linear maps

4.7

Tensor powers

N from k V to R. The identification is such as to send a decomposable element N N l1 . . . lk to the map that sends any given k-tuple (v1, . . . , vk) to l1(v1) . . . lk(vk). There is an analogous way to view the k-fold N tensor power of the dual of a given bundle p: E ! M. Indeed, a section k E* defines a k-linear, fiber L L L L preserving map from E . . . E to M  R. Conversely, a map f : E . . . E !N M  R that preserves the fibers and is linear in each entry defines a section of k E*.

4.7.1

Symmetric powers

A k-bilinear map, f , from k V to R is said to be symmetric if f ðv1 ; …; vi ; …; vj ; …; vk Þ ¼ f ðv1 ; …; vj ; …; vi ; …; vk Þ for any pair of indices i, j 2 {1, . . . , k} and vectors (v1, . . . , vk) 2 k V. The vector N space of symmetric, k-bilinear maps from kV to R is a vector subspace of kV*, this denoted by Symk(V*). Note that this vector space is isomorphic to the vector space of homogeneous, k’th order polynomials on V. For example, an isomorphism sends a symmetric map f to the polynomial v ! f (v, . . . , v). The inverse of this isomorphism sends a homogeneous polynomial ℘ to the symmetric, k-bilinear map that sends a given (v1, . . . , vk) 2 kV to ð@t@1 @t@k ℘ðt1 v1 þ þ tk vk ÞÞjt1 ¼ ¼ tk ¼0 : By the way, a smooth map f : V ! R such that f (rv) ¼ rkf (v) for all r 2 R and v 2 V is, de facto, a homogeneous polynomial of degree k. Indeed, any such map must 1 @k obey f ðvÞ ¼ k! ð@tk f ðtvÞÞjt¼0 ; and so is given by the order k term in f ’s Taylor’s expansion at the origin in V. Let p: E !N M again denote a vector bundle. The bundle Symk(E*) is the subbundle of k E* whose restriction to any given fiber defines a symmetric, k-linear function on the fiber. Said differently, the fiber of Symk(E*) over any given p 2 M is the vector space of maps from Ejp to R that obey f (rv) ¼ rkf (v) for each r 2 R and v 2 Ejp. A standard convention sets Sym0(E) ¼ M  R.

4.7.2

Antisymmetric powers

Fix k 2 {1, . . . n}. A k-linear map from k V to R is said to be antisymmetric when f ðv1 ; …; vi ; …vj ; …; vk Þ ¼ f ðv1 ; …; vj ; …vi ; …; vk Þ for any pair of indices i, j 2 {1, . . . , k} and vectors (v1, . . . , vk) 2 kV. The vector space of antisymmetric, k-linear maps from k V to R is also a vector subspace

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N

this denoted by ∧kV*. Note that ∧kV* ¼ {0} if k > n, and that n! ∧ V* ¼ R. If k 2 {1, . . . , n}, then ∧kV* has dimension k!ðnkÞ! . Suppose that {v1, . . . , vn} is a given basis for V. Introduce {l1, . . . , ln} to denote the dual basis for V*. This set determines a basis of sections of ∧kV*, where any given basis element is labeled by a k-tuple (a1, . . . , ak) of integers that obey 1  a1 < . . . < ak  n. The corresponding basis element is written as of

k V*,

n

la1 ∧la2 ∧ . . . ∧ lak ; this corresponds to the antisymmetric map from k V to R that assigns 1 to the element (va1 ; . . .; vak ) and zero to basis elements that have an entry whose label is not from {a1, . . . , ak}. Note that there is a canonical homomorphism from N 0 0 ∧kV* ∧k V* to ∧kþk V* given by antisymmetrization. The image of a pair (o, m) under this homomorphism is written as o ∧ m. This homomorphism is sometimes called the wedge product and other times the exterior product. The 0 wedge/exterior product is such that o ∧ m ¼ (1)kk m ∧ o. Let p: E ! M denote a given N vector bundle. If k 2 {1, . . . , n}, the bundle ∧k E* is the subbundle in k E* whose fiber over any given point p 2 M consists of the elements that define antisymmetric, k-linear functions on Ejp. The bundle ∧k E* is the 0-dimensional bundle M  {0} if k > n. Meanwhile ∧nE* is a real line bundle. However, it need not be isomorphic to the trivial line bundle! To say more about the latter, suppose that U  M is an open set and that jU : EjU ! U  Rn is a local trivialization of E over U. Let U0  M denote a second open set, and suppose that jU0 : EjU0 ! U0  Rn is a local trivialization over U0 . Assume that U0 U 6¼  so that there is a well-defined bundle transition function gU0 ,U: U U0 ! Gl(n; R). The corresponding bundle transition function for ∧n E* is det(gU0 ,U). The bundle ∧nE is denoted by det(E). The exterior/wedge product applied fiber-wise defines a bundle homomorN 0 0 phism from ∧kE* ∧k E* to ∧kþk E* which is also called the exterior or wedge product. Of special interest is the case of when E ¼ TM in which case a section of ∧kT*M is called a k-form. ¨

¨

46

Additional reading  Analysis and Algebra on Differentiable Manifolds, P. M. Gadea and J. Munoz, Springer, 2001.  Introduction to Differential Topology, T. Bro¨cker and K. Ja¨nich, Cambridge University Press, 1972.

Additional reading

 Tensor Analysis on Manifolds, Richard L. Bishop and Samuel Goldberg, Dover Publications, 1980.  K-Theory, An Introduction, Max Karoubi, Springer, 1978.  Introduction to Smooth Manifolds, John M. Lee, Springer, 2003.  Fiber Bundles, Dale Husemo¨ller, Springer, 1993.

47

5

Maps and vector bundles

Suppose that M, N are smooth manifolds. As explained in what follows, any given map c: M ! N can be used to associate a vector bundle on M to one on N.

5.1

The pull-back construction

Let p: E ! N denote a vector bundle with fiber dimension n. There is a canonical bundle, denoted c *E, over M, also with fiber dimension n, which comes with a canonical map cÐ : c*E ! E that restricts to the fiber over any given point p 2 M as a vector space isomorphism from (c*E)jp to Ejc(p). The bundle c*E is the submanifold in M  E of pairs {(m, e) : c(m) ¼ p(e)}. This is to say that the fiber of c*E over m 2 M is the fiber of E over c(m). The map cÐ sends a pair (m, e) 2 c*E (thus m 2 M and e 2 N with p(e) ¼ c(m)) to the point cÐ (m, e) ¼ e. To be somewhat more explicit, suppose that U  N is an open set and that jU : EjU ! U  Rn is a bundle isomorphism. Then the c*E is isomorphic over c1(U)  M to c1(U)  Rn, the isomorphism taking a pair (m, e) to the pair (m, lU(e)) where lU : EjU ! Rn is defined by writing jU(e) as the pair (p(e), lU(e)). Let U0  N denote a second open set with nonempty intersection with U and with a corresponding bundle trivialization jU0 : EjU0 ! U0  Rn. The associated transition function gU0 ,U : U0 U ! Gl(n; R) determines the transition function of c*E for the sets c1(U0 ) c1(U) as the latter is gU0 ,U  c. Sections of bundles also pull-back. To elaborate, suppose that p: E ! N is a smooth bundle and s: N ! E is a section. Let c: M ! N denote a smooth map. Then s  c defines a section of c*E. ¨ ¨

Example 5.1 Let S  R3 denote a surface. Define a map from S to S2 by associating to each point p 2 S the vector normal to S (pointing out from the region that is enclosed by S). This map is called the Gauss map. Denote it by n. Then TS is isomorphic to n*TS2. The isomorphism is obtained as follows: View TS2  R3  R3 as the set of pairs of the form (v, w) with jvj2 ¼ 1 and v·w ¼ 0. Here, v·w denotes the dot product in R3. Granted this, then n*TS2 can be

5.2

Pull-backs and Grassmannians

viewed as the set {(p, w) 2 S  R3 : n(p)·w ¼ 0}. As explained in Chapter 2.4, this is identical to TS. A concrete example is had using a surface of genus g as constructed in Chapter 1.3 from a function, h, on R2. Recall that the surface is given by the set of points in R3 where z2  h(x, y) ¼ 0. The corresponding Gauss map is given by  , ∂ ∂ h; 2z ðjdhj2 þ 4z2 Þ1=2 : njðx;y;zÞ ¼  h; ∂x ∂y Example 5.2 Pick a point q in the region of R3 bounded by S and define a map, c, from S to S2 by taking the direction from the given point on S to q. Is c*TS2 isomorphic to TS? What if the point q is outside S instead of inside? Example 5.3 Let M denote a given m-dimensional manifold. What follows constructs a map from M to Sm that is relevant to one of the constructions in Chapter 3.3.5. To begin the story, fix a finite set   M. Assign to each point p 2  a coordinate chart (U, jp : U ! Rm) with p 2 U such that jp(p) ¼ 0 2 Rm. Fix r > 0 and small enough so that the open ball, B, of radius r about the origin in Rm is in the image of each p 2  version of jp; and so that the collection {Up ¼ j1 p (B)}p2 consists of pair-wise disjoint sets. Use this data to define a map, c, from M to Sm  Rmþ1 as follows: The map c send all of the set M( p Up) to the point (0, . . . , 0, 1). To define c on a given version of Up, write Rmþ1 ¼ Rm  R. Introduce from Appendix 1.2 the function w and set wp : Up ! [0, 1] to equal w(jjpj). This done, the map c sends a p 2 Up to the point (2wpjp, wp2  jjpj2)/ (wp2 þ jjpj2)1. To continue, fix a positive integer n and a map g: Sm1 ! Gl(n; R). Define a rank n vector bundle E ! Sm by using the coordinate patches Uþ ¼ Sm(0, . . . , 1) and U– ¼ Sm(0, . . . , þ1) with the transition function that gU–,Uþ send a point (x, xnþ1) 2 Sn  Rn  R to g(x /jxj). The resulting bundle c*E ! M is isomorphic to the bundle constructed in Chapter 3.3.5 using the data consisting of  and the set {(Up, jp, gp ¼ g)}p2.

¨

5.2

Pull-backs and Grassmannians

As it turns out, any given vector bundle is isomorphic to the pull-back of a tautological bundle over a Grassmannian, these as described in Chapter 3.3.4. The next proposition makes this precise. Proposition 5.4 Let M denote a smooth manifold, let n denote a positive integer, and let p: E ! M denote a given rank n vector bundle. If m is sufficiently

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large, there exists a map cm : M ! Gr(m; n) and an isomorphism between E and the pull-back via cm of the tautological bundle over Gr(m; n). Proof of Proposition 5.4 As noted in Chapter 4.1, the bundle E is isomorphic to a subbundle of some N 1 version of the product bundle M  RN. Use an isomorphism of this sort to view E as a subbundle in just such a product bundle. Let l: M  RN ! RN denote the projection map. The assignment to x 2 M of the n-dimensional vector subspace l(Ejx)  RN defines a smooth map c: M ! Gr(N; n). Let EN ! Gr(N; n) denote the tautological Rn-bundle. This is to say that the fiber of EN over a given subspace  2 Gr(N; m) is the vector space . Thus, the fiber of EN over c(x) is Ejx and so E is the pull-back via c of the tautological bundle.

5.3

Pull-back of differential forms and push-forward of vector fields

There is a canonical, vector space homomorphism that maps C1(N;∧kT*N) to C1(M; ∧kT*M) for any k 2 {0, 1, . . . , }, this is also denoted by c*. It is defined so as to commute with wedge product, exterior multiplication and to factor with regards to compositions. This map is defined by the following rules  c*f ¼ f  c and c*df ¼ d(c*f) when f 2 C1(N) ¼ C1(N; ∧0 T*N).  c*(fo) ¼ (c*f) c*o when f 2 C1(N) and o 2 C1(N; ∧kT*N) for any k. Induction on k using these two rules defines c* as an algebra homomorphism L L from k0 C1(N; ∧kT*N) to k0 C1(M; ∧kT*M) that commutes with d. Moreover, if Z is a third manifold and j: N ! Z is a smooth map, then (j  c)* ¼ j*  c*. The homomorphism c* is called the pull-back homomorphism, and c*o is said to be the pull-back of o. The picture of the pull-back homomorphism in local coordinates shows that the latter is a fancy version of the Chain rule. To elaborate, fix positive integers m and n; and let f denote a smooth map between Rm and Rn. Introduce coordinates (x1, . . . , xm) for Rm and coordinates (y1, . . . , yn) for Rn. Use (f1, . . . , fn) to denote the components of the map f. Then   @fk  f*ðdyk Þ ¼ ∑lim  dxi : @xi x Note that d(f*dyk) ¼ 0 because indices i and j.

@2 f @xi @xj

f ¼ @x@j @x for any given function f and i 2

5.3

Pull-back of differential forms and push-forward of vector fields

Use of local coordinate charts gives a picture of this map c* which is essentially that given above for the cases M ¼ Rm and N ¼ Rn. To elaborate, let M and N again be smooth manifolds and c: M ! N a smooth map. Let U  M denote a coordinate chart with coordinate map jU : U ! Rm=dim(M). Let (x1, . . . , xm) denote the Euclidean coordinate functions on Rm, and recall from Chapter 3.9 that the map jU can be used to identify the basis of sections {dxi}1im of T*Rm with a corresponding basis of sections of T*MjU. Use the notation {dxUi}1im to denote the latter. Let V  N denote a coordinate chart whose intersection with c(U) is not empty, and let jV : V ! Rn=dim(N) denote the coordinate map. Use (y1, . . . , yn) to denote the Euclidean coordinates of Rn and use fV to define the basis of sections {dyV i }1in for T*NjV. The pull-back via c of a given dyVk is given by   @fk  ðc*dyVk Þjx ¼ ∑1im  ðdxUi Þjx @xi j ðxÞ U

1

where f here denotes jU  c  (jV) , this is a map from a domain in Rm to one in Rn. By the way, this pull-back map c* is closely related to the previously defined notion of the pull-back bundle. Indeed, c* can be defined as the composition of two linear maps. The first map pulls back a section of ∧kT*N to give a section of c*(∧kT*N) over M. This as described in Chapter 5.1 above. The second is induced by a certain canonical vector bundle homomorphism from c*(∧kT*N) to ∧kT*M. To say more about the latter, let V  N and jV: V ! Rn=dim(N) be as described in the preceding paragraph. Then the sections {(dyVi )  c}1in give a basis of sections for (c*T*N)jV. Now suppose that U  M and jU : U ! Rm=dim(M) is also as described in the preceding paragraph. Then the aforementioned bundle homomorphism from c*T*N to T*M restricts over U so as to send any given ((dyVi)  c)jx to what is written on the left-hand side of the equation in the preceding paragraph. There is a dual of sorts to the pull-back map c*: C1(N; T*N) ! 1 C (M; T*M), this canonical push-forward map from TM to TN that restricts to any given x 2 M version of TMjx as a linear map to TNjc(x). This pushforward is denoted by c*. To elaborate, suppose that v 2 TM and o is a section of T*N. The pairing between o and c*v is, by definition, equal to that between the pull-back c*o and v. Note that this push-forward can be viewed as the composition of a bundle homomorphism from TM to c*TN followed by the map cÐ : c*TN ! TN. The former is the dual homomorphism to that described in the preceding paragraph from c*T*N to T*M. The next section of this chapter describes some important examples of pullback and push-forward.

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5.4

Invariant forms and vector fields on Lie groups

Let G denote a Lie group. Then any given element g 2 G defines two maps from G to itself, these being the maps that sent m 2 G to lg(m) ¼ gm and to rg(m) ¼ mg. The first is called left translation by g, and the second is called right translation by g. Both are diffeomorphism with inverses given respectively by left and right translation by g1. A 1-form o on G is said to be left invariant if lg* o ¼ o for all g 2 G. The 1-form o is said to be right invariant if rg* o ¼ o for all g 2 G. By the same token, a vector field v on G is left invariant or right invariant if lg* v ¼ v or rg* v ¼ v for all g 2 G as the case may be. Suppose that o is a left invariant 1-form. If o is nonzero on TGji, then o is nowhere zero. This follows by virtue of the fact that lg is a diffeomorphism. This implies that the restriction map from C1(G; T*G) to T*Gji identifies the vector space of left-invariant 1-forms with T*Gji. This said, then a basis for T*Gji gives a basis for the left-invariant 1-forms on G, and this basis gives a basis of sections for T*G over the whole of G. This then supplies a vector bundle isomorphism from the product bundle G  T*Gji to T*G. By the same token, the vector space of right-invariant 1-forms on G is also isomorphic to T*Gji and so a basis for T*Gji extends to the whole of G as a basis of rightinvariant 1-forms on G, which defines a basis of sections for T*G. This then supplies a second isomorphism between G  T*Gji and T*G. These two isomorphisms will not, in general, agree. Of course, a similar thing can be said with regards to TG and the vector spaces of left- and right-invariant vector fields. A basis for TGji extends as a basis of left-invariant vector fields, and also as a basis of right-invariant vector fields. Either basis gives a basis of sections of TG over the whole of G and thus an isomorphism from the product vector bundle G  TGji to TG. Example 5.5 Introduce the n2-dimensional Euclidean space M(n; R) of n  n real matrices. In what follows, I use notation whereby the entries {mij}1i,jn of a given matrix define the Euclidean coordinates on M(n; R). This understood, the n2 differentials {dmij}1i,jn give a basis of sections of T*M(n; R). Let i: Gl(n R) ! M(n; R) denote the tautological embedding as the group of matrices with nonzero determinant. Fix a nonzero matrix q and define the 1-form oq on Gl(n; R) by the rule oq jm ¼ trðqm1 dmÞ: P To review the notation, what is written above is oq ¼ 1i,j,kn (qij(m1)jkdmki.) This 1-form is left invariant: Indeed, let g 2Gl(n; R). P P Then lg*(mij) ¼ 1kn gikmki and so lg*(dmij) ¼ 1kn gikdmkj. Thus,

5.5

The exponential map on a matrix group

(lg*oq)jm ¼ tr(qm1g1gdm) ¼ oqjm. The form oq is not right invariant as can be seen from the formula (rg*o)jm ¼ tr(gqg1m1dm), this being og1 qg . A similar calculation finds that the form tr(m1qdm) is right invariant but not left invariant. The form oq at the identity i 2 Gl(n; R) is nonzero since q is nonzero, and so the form oq is nowhere zero on Gl(n; R). To see what a left-invariant vector field looks like, fix again some matrix q. This done, introduce   ∂ ∂ : ¼ ∑1i;j;kn qji mkj vq jm ¼ tr qmT ∂m ∂mki Any such vector field is left invariant, and any left-invariant vector field can be written in this way.

5.5

The exponential map on a matrix group

Recall that the space M(n; R) of n n real valued matrices is a Euclidean space with the coordinate functions given by the n2 entries of a matrix. Fix an n  n matrix m 2 M(n; R). Use this matrix to define the exponential map em: M(n; R) ! M(n; R) by the rule em(a) ¼ m ea, where 1 1 e a ¼ i þ a þ a2 þ a 3 þ . . . 2 3! is the power series with the coefficient of ak equal to 1/k!. This sequence converges absolutely and so defines a smooth map from M(n; R) to itself. Note that the matrix ei(a) is invertible, this by virtue of the fact that ei(a)1 ¼ ei(a). Thus em(a) 2 Gl(n; R) if m is. In particular, the map ei maps M(n; R) into Gl(n; R). As the differential of the map a ! ea at a ¼ 0 is the identity map on M(n; R), it follows as a consequence of the inverse function theorem that this restricts to some ball in M(n; R) as a diffeomorphism onto some neighborhood of i in Gl(n; R). With the preceding understood, fix q 2 M(n; R) and reintroduce the leftinvariant 1-form oq ¼ tr(q m1dm) on Gl(n; R). The pull-back of this form by ei is the 1-form ð1 e*i oq ¼ ds trðesa qesa daÞ: 0

To elaborate, view Gl(n; R) as an open subset of M(n; R) so as to identify TGl(n; R)ji with M(n; R). Granted this identification, then e* i oq is neither more

53

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nor less than a linear functional on M(n; R). This linear functional maps any given matrix c to ð1 ds trðesa qesa cÞ: 0

What follows considers the exponential map in the context of the groups Sl(n; R) and SO(n).

5.5.1

The group Sl(n; R)

Recall that group Sl(n; R) sits in Gl(n; R) as the subgroup of matrices with determinant 1. As it turns out, ei maps the vector space of trace zero matrices in M(n; R) into Sl(n; R). Indeed, this follows from the fact that  d   ln det ei ðsaÞ ¼ trðaÞ: ds As noted in Chapter 3.5, the tangent space to Sl(n; R) at the identity is the vector space of trace zero, n  n matrices. Given that the differential of ei at 0 is the identity map on M(n; R), it follows directly from the inverse function theorem that ei maps a ball about the origin in the vector space of trace zero, n  n matrices diffeomorphically onto an open neighborhood of the identity in Sl(n; R). Likewise, em for m 2 Sl(n; R) maps this same ball diffeomorphically onto an open neighborhood of m in Sl(n; R). Let i: Sl(n; R) ! Gl(n; R) denote the tautological inclusion map, and let q denote a given n  n matrix. Then i*oq is nonzero unless q is a multiple of the identity, in which case i*oq ¼ 0. The fact that the pull-back of the q ¼ i version of oq is zero on TSl(n; R)ji can be seen using the formula given above for its pull-back via ei. As i*oi is a left-invariant 1-form on Sl(n; R), its vanishing on TSl(n; R)ji implies that it must vanish everywhere.

5.5.2

The group SO(n)

Let A(n; R) again denote the vector space of n  n antisymmetric matrices. This is to say that a 2 A(n; R) when a’s transpose obeys aT ¼ a. The map ei as depicted above obeys ei(a)T ¼ ei(aT) and so eiTei ¼ 1 when a 2 A(n; R). It follows from this that ei restricts to A(n; R) so as to map the letter subspace into SO(n). To say more, recall that Chapter 3.5 identifies TSO(n)ji with A(n; R). Granted this identification, and granted that the differential at a ¼ 0 of ei is the identity map, it follows from the implicit function theorem that ei maps a ball in A(n; R) diffeomorphically onto a neighborhood of i 2 SO(n). By the same token

5.6

The exponential map and right/left invariance on Gl(n; C) and its subgroups

if m is any given matrix in SO(n), then the map em ¼ mei restricts to this same ball as a diffeomorphism onto a neighborhood of m in SO(n). Let q again be a matrix in M(n; R). Let i: SO(n) ! Gl(n; R) denote the tautological inclusion map. As explained next, the form i*oq is zero if qT ¼ q and never zero if qT ¼ q unless q ¼ 0. To see that i*oq ¼ 0 when q is symmetric, consider its pull-back via the exponential map ei as depicted above. For any given a 2 A(n; R), this defines a linear form on A(n; R) ¼ TSO(n)ji whose Ð1 pairing with c 2 A(n; R) is 0 ds trðesa qesa cÞ: The term in the integrand is, for any given value of s 2 [0, 1], the trace of the product of a symmetric matrix with an antisymmetric matrix. Such a trace is always zero. To see that i*oq 6¼ 0 when q 2 A(n; R), it is enough to prove it nonzero on TSO(n)ji since this form is invariant under left multiplication. The formula above shows that i*oq at the identity pairs with c 2 A(n; R) to give tr(qc). Taking c ¼ –q ¼ qT shows that i*oqji is nonzero on A(n; R) unless q is zero. Given that i*oq is left invariant, this implies that any q 6¼ 0 version is nowhere zero on SO(n).

5.6

The exponential map and right/left invariance on Gl(n; C) and its subgroups

The group Gl(n; C) is described in Chapter 2.4. Recall that there are two descriptions. The first views Gl(n; C) as the open subset in the Euclidean space M(n; C) of n  n matrices with complex entries that consists of the matrices with nonzero determinant. As a manifold, M(n; C) has (real) dimension 2n2, and its Euclidean coordinates are taken to be the real and imaginary parts of the entries of a given matrix m. This understood, view dm as a matrix of C-valued differential forms on M(n; C) whose real and imaginary parts supply a basis of sections for T*M(n; C). With the preceding understood, let q 2 M(n; C). Then the C-valued differential form oq ¼ tr(q m1dm) on Gl(n; C) is left invariant, and any C-valued, leftinvariant 1-form on the group Gl(n; C) can be written in this way, and any R-valued left invariant 1-form is the real part of some oq. By the same token, the C-valued 1-form tr(m1q dm) is right-invariant, and any C-valued, rightinvariant 1-form on the group Gl(n; C) is of this sort. Moreover, any R-valued right invariant 1-form is given by the real part of some oq. The C-valued, right- and left-invariant vector fields on Gl(n; C) are given @ @ respectively by the rules trðq mT @m Þ and trðmT q @m Þ where the notation is as @ follows: What is written as @m denotes here the n  n matrix whose (k, j) entry is by definition @m@ kj ¼ 12 ð@x@kj  i @y@ Þ with xkj and ykj denoting the respective real kj

and imaginary parts of the complex coordinate function mkj.

55

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5 : Maps and vector bundles

Fix m 2 M(n; C) and define the exponential map em: M(n; C) ! M(n; C) using the same formula as used to define the M(n; R) version. As in the case of M(n; R), the map em sends M(n; C) into Gl(n; C) when m 2 Gl(n; C), and it restricts to some m-independent ball about the origin in M(n; C) as a diffeomorphism onto an open neighborhood of m. It is also the case that em sends the complex, codimension 1 vector space of traceless matrices in M(n; C) into Sl(n; C) when m 2 Sl(n; C); and it restricts to a ball about the origin in this subspace as a diffeomorphism onto an open neighborhood of m in Sl(n; C). Likewise, em maps the vector subspace A(n; C)  M(n; C) of anti-Hermitian matrices into U(n) when m 2 U(n) and it restricts to some m-independent ball about the origin in A(n; C) as a diffeomorphism onto a neighborhood of m in U(n). Finally, this same em maps the vector subspace A0(n; C)  A(n; C) of trace zero matrices into SU(n) when m 2 SU(n); and it restricts to some m-independent ball about the origin in A0(n; C) as a diffeomorphism onto an open neighborhood of m in SU(n). The proofs of all of these assertions are identical to the proofs of their analogs given in the preceding section of this chapter. Let i denote the inclusion map from any of Sl(n; C), U(n) or SU(n) into Gl(n; C). In the case of Sl(n; C), the pull-back i*oq is zero if q ¼ ci with c 2 C and nowhere zero otherwise. The form i*oq on Sl(n; C) is a left-invariant, C-valued 1-form, and any such 1-form can be written as i*oq with tr(q) ¼ 0. Any R-valued, left-invariant 1-form is the real part of some i*oq. In the case of U(n), the 1-form i*oq is zero if q{ ¼ q, and it is nowhere zero if q{ ¼ q. Any such form is left-invariant, and any C-valued, left-invariant 1-form on U(n) is of this sort. Likewise, an R-valued, left-invariant 1-form can be written as the real part of some oq. In the case of SU(n), the pull-back i*oq is also zero if q is Hermitian, or if q ¼ iri with r 2 R; and it is nowhere zero if q{ ¼ iq and tr(q) ¼ 0. Any real valued, left-invariant 1-form is the real part of some i*oq of this sort. The proofs of all the preceding statements are essentially identical to the proofs of the analogous statements in the preceding section of this chapter. As noted at the outset, Chapter 2 provides two views of Gl(n; C). The second view regards M(n; C)  M(n; 2R) as the subvector space over R of matrices that commute with a certain matrix, j0, whose square is i. The latter subvector space is denoted by Mj. Glj  Gl(2n; R) denote the latter’s intersection with Mj. Let a ! ei(a) denote the exponential map defined on M(n; 2R). As can be seen from the formula, this map has the property that j0 ei(a) j0 ¼ ei(j0aj0). As a consequence, the M(n; 2R) version of em restricts to Mj so as to map the latter vector space to the subgroup Glj when m 2 Mj. The identification of Mj with M(n; C) that is obtained by writing m 2 Mj as a linear combination of the eigenvalue þi eigenvectors of j0 and their complex conjugates identifies the restriction to Mj of this M(2n; R) version em with the M(n; C) version of em. This understood, all of the assertions given so far in this section that concern the

5.7

Immersion, submersion and transversality

exponential map on Gl(n; C) and its subgroups can be reproved without recourse to complex numbers by considering the M(2n; R) version of em on Mj. The identification between Mj and M(n; C) and between Glj and Gl(n; C) can also be used to prove the assertions about left- and right-invariant 1-forms and vector fields without recourse to complex numbers.

5.7

Immersion, submersion and transversality

This section of the chapter uses the notion of vector push-forward to reinterpret and then elaborate on what is said in Chapter 1.4. To set the stage, suppose that M and Y are smooth manifolds, and suppose that c: Y ! M is a smooth map. Chapter 1.4 introduced the descriptives immersion and submersion as applied to c. These notions can be rephrased in terms of the push-forward c* as follows: Definition 5.6 The map c is respectively an immersion or submersion when the vector bundle homomorphism c*: TY ! c*TM is injective on each fiber or surjective on each fiber. A useful notion that is intermediate to the immersion and submersion cases is described next. To set this stage, suppose that Y, M and c are as described above. Suppose in addition that Z  Y is a submanifold. The map c is said to be transversal to Z when the following condition is met: Let y 2 Y denote any given point with c(y) 2 Z. Then any vector v 2 TMjc(y) can be written as c*vY þ vZ with vY 2 TYjy and vZ 2 TZjc(y). The proposition that follows applies the notion of transversality to construct a submanifold of Y. This proposition uses n, m and d for the respective dimensions of Y, M and Z. Proposition 5.7 Let M, Y and Z  M be as just described, and suppose that c: Y ! M is transversal to Z. Then c1(Z) is a smooth submanifold of Y of dimension n þ d  m. The proof of this proposition is given momentarily. Note in the meantime that when Z is a point, then c is transversal to Z if and only if c* is surjective at each point in c1(z). Proof of Proposition 5.7 Let y denote a given point in c1(Z). Fix an open neighborhood, U  Y, of y with a coordinate chart map j: U ! Rn that sends y to the origin. Fix, in addition, an open neighborhood, V  M, of c(y) with a coordinate chart map j0 : V ! Rm that sends c(y) to the origin and sends Z V to the d-dimensional subspace of points (u1, . . . , um) 2 Rm with udþ1 ¼ . . . ¼ um ¼ 0.

57

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5 : Maps and vector bundles

Chapter 1.3 says something about such a coordinate chart map. Let f ¼ (f 1, . . . , f m) denote the map from Rn to Rm given by j0  c  j1. Let f denote the map from a neighborhood of the origin in Rn to Rmd given by x ! fðxÞ ¼ ðfdþ1 ðyÞ; . . . ; fm ðyÞÞ: The transversality assumption guarantees that f is a submersion near the origin in Rn. As a consequence, f1(0) is a submanifold near the origin in Rn. This submanifold is, by construction, j(c1(Z)).

Additional reading  Differential Forms in Algebraic Topology, Raoul Bott and Loring W. Tu, Springer, 1982.  Manifolds and Differential Geometry, Jeffrey M. Lee, American Mathematical Society, 2009.  Geometry of Differential Forms, Shigeyuki Morita, American Mathematical Society, 2001.  An Introduction to Manifolds, Loring W. Tu, Springer, 2008.  Introduction to Smooth Manifolds, John M. Lee, Springer, 2003.  Differential Topology, Victor Guillemin and Alan Pollack, Prentice-Hall, 1974.  A Geometric Approach to Differential Forms, David Bachman, Birkhauser, 2006.

6

Vector bundles with Cn as fiber

Just as there are vector spaces over C, there are vector bundles whose fibers can be consistently viewed as Cn for some n. This chapter first defines these objects, and then supplies a number of examples.

6.1

Definitions

What follows directly are three equivalent definitions of a complex vector bundle. The first definition highlights the fact that a complex vector bundle is a vector bundle of the sort introduced in Chapter 3.1 with some extra structure. Definition 6.1 A complex vector bundle of rank n is a vector bundle, p: E ! M, in the sense defined in Chapter 3.1 with fiber dimension 2n, but equipped with a bundle endomorphism j: E ! E such that j2 ¼ 1. This j allows one to define an action of C on E such that the real numbers in C act as R on the underlying real bundle of fiber dimension 2n; and such that the number i acts as j. Doing this identifies each fiber of E with Cn up to the action of Gl(n; C). The endomorphism j is said to be an almost complex structure for E. The underlying real bundle with fiber R2n is denoted by ER when a distinction between the two is germaine at any given time. The preceding definition highlights the underlying real bundle structure. There are two equivalent definitions that put the complex numbers to the fore. Definition 6.2 This definition gives the C analog of what is given in Chapter 3.1: A complex vector bundle E over M of fiber dimension n is a smooth manifold with the following additional structure:  A smooth map p: E ! M.  A smooth map oˆ: M ! E such that p oˆ is the identity.  A smooth map m: C  E ! E such that

6 : Vector bundles with Cn as fiber

a) p(m(c, v)) ¼ p(v) b) m(c, m(c0 , v)) ¼ m(cc0 , v) c) m(1, v) ¼ v d) m(c, v) ¼ v for c 6¼ 1 if and only if v 2 im(oˆ).  Let p 2 M. There is a neighborhood, U  M, of p and a map lU : p1(U) ! Cn such that lU : p1(x) ! Cn for each x 2 U is a diffeomorphism obeying lU(m(c, v)) ¼ c lU(v). Definition 6.3 This definition gives the C analog of the cocycle definition from Chapter 3.2: A complex vector bundle of rank n  1 is given by a locally finite open cover U of M with the following additional data: An assignment to each pair U, V 2 U a map gU,V: U V ! Gl(n; C) such that gU;V ¼ g1 V;U with the constraint that if U, V, W 2 U, then gU,V gV,W gW,U ¼ i on U V W. Given this data, define the bundle E to be the quotient of the disjoint union U2U (U  Cn) by the equivalence relation that puts (p0 , v0 ) 2 U0  Cn equivalent to (p, v) 2 U  Cn if and only if p ¼ p0 and v0 ¼ gU0 ,U(p)v. The constraint involving the overlap of three sets guarantees that this does, indeed, specify an equivalence relation. ¨

¨



¨

60

6.2

Comparing definitions

The argument to equate the Definitions 6.2 and 6.3 is almost verbatim what is given in Chapter 3.2 that equates the latter’s cocycle definition of a vector bundle with the definition given in Chapter 3.1. Indeed, one need only change R to C in Chapter 3.2’s argument. To compare Definitions 6.1 and 6.2, suppose first that E is a complex vector bundle in the sense of Definition 6.2. The needed data from Chapter 3.1 for the underlying real bundle ER consist of a projection map p: ER ! M, a zero section oˆ: M ! ER, a multiplication map m: R  E ! E. In addition, each p 2 M should be contained in an open set U which comes with a certain map lU : ERjU ! R2n. The ER is, first of all, declared equal to E. Chapter 3.1’s projection p is the projection given in Definition 6.2. Likewise, Chapter 3.1’s zero section oˆ is that given in Definition 6.2. Meanwhile, Chapter 3.1’s map m from R  ER to ER is obtained from Definition 6.2’s map m by restricting the latter to the real numbers R  C. Granted the preceding, any given version of Chapter 3.1’s map lU can be taken equal to the corresponding version of Definition 6.2’s map lU after identifying Cn as R2n by writing the complex coordinates (z1, . . . , zn) of Cn in terms of their real and imaginary parts to obtain the Euclidean coordinates for n R2 ¼ R2n. With ER so defined, Definition 6.1 requires only the endomorphism j. The latter is defined by the rule j(v) ¼ m(i, v) with m here given by Definition 6.2.

6.2

Comparing definitions

Now suppose that E is a complex bundle according to the criteria from Definition 6.1. The data needed for Definition 6.2 are obtained as follows: First, Definition 6.2’s projection p and Definition 6.2’s zero section oˆ are those from the underlying real bundle ER. Definition 6.2’s map m is defined as follows: Write a given complex number z in terms of its real and imaginary parts as z ¼ a þ ib. Let v 2 E. Then m(a þ ib, v) is defined to be av þ bjv. It remains only to specify, for each p 2 M, an open set, U, containing p with a suitable map lU : EjU ! Cn. This last task requires a preliminary lemma. To set the stage for the required lemma, reintroduce the standard almost complex structure j0 from Chapter 2.4. Lemma 6.4 Let p: ER ! M denote a real vector bundle with fiber dimension 2n with an almost complex structure j: E ! E. There is a locally finite cover U for M of the following sort: Each set U 2 U comes with an isomorphism jU : ERjU ! U  R2n that obeys jU  jj1 U ¼ j0 at each point of U. This lemma is proved momentarily. It has the following consequence: Let U denote a cover of the sort given by the lemma. Let U  U, and set lU;R : EjU ! R2n equal to the composition of jU followed by the projection to the R2n factor. Define the complex coordinates (z1, . . . , zn) for Cn from the real coordinates (x1, . . . , x2n) of R2n by writing zk ¼ x2k1 þ ix2k. Doing so allows lU;R to be viewed as a map, lU, from EjU to Cn. Because jU intertwines j with j0, the latter map is equivariant with respect to the action of C on E and C on Cn. And, it restricts to any given fiber as an isomorphism between complex vector spaces. Proof of Lemma 6.4 Fix p 2 M and an open set V  M that contains p with an isomorphism cV : ERjV ! V  R2n. Use jV to denote cV j cV1. At each point of V, this endomorphism of R2n defines an almost complex structure on R2n. The plan in what follows is to find a smooth map h, from a neighborhood of p to Gl(2n; R) such that h  jV  h1 ¼ j0. Given such a map, set U to be this same neighborhood and define jU at any given point in U to be h cV. To obtain h, digress momentarily and suppose that j1 is any given almost complex structure. Then there exists g 2 Gl(2n; R) such that gj1g1 ¼ j0. Indeed, this follows by virtue of the fact that j1 and j0 are diagonalizable and have eigenvalues only i, each with multiplicity n. To elaborate, fix a basis of eigenvectors for j1 with eigenvalue þi and likewise a basis for j0. Define g by the rule that sends the respective real and imaginary parts of the k’th basis vector for j1 to those of the corresponding basis vector for j0. Granted the preceding, it follows that cV can be changed by cV ! g cV with g 2 Gl(2n; R) so that the resulting version of jV is equal to j0 at p. With the preceding understood, reintroduce the exponential map a ! ea as defined in Chapter 5.5, and recall that the latter restricts to some ball about the origin in M(2n; R) as a diffeomorphism onto a neighborhood of i in Gl(2n; R).

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Granted the latter fact, it then follows that jV near p can be written as j0ea where a is a smooth map from a neighborhood of p in V to M(2n; R). Moreover, a must be such that ej0 aj0 ea ¼ i; for this condition is necessary and sufficient for j2V to equal i. Because (ea)1 ¼ ea, this last condition requires that j0 a j0 ¼ a on some neighborhood of p. With this last point in mind, use the power series definition of ea to write the latter ea ¼ ea/2ea/2, and then write jV ¼ j0 ea=2 ea=2 ¼ ea=2 j0 ea=2 : Then take h ¼ ea/2.

6.3

Examples: The complexification

The simplest example of a complex vector bundle is the product bundle M  Cn. The underlying rank 2n real bundle, ER, is M  R2n; the almost complex structure is j0. The examples given next are not much more complicated. These examples start with some given vector bundle p: E ! M with fiber Rn. The resulting complex bundle has fiber Cn; it is denoted EC and is said to be the complexification of E. The bundle EC is defined so that its fiber of EC at any N given p 2 M is canonically isomorphic to C R Ejp as a complex vector space. What follows describes EC in the context of Definition 6.1. N E, this a bundle To start, introduce the tensor product bundle (M  R2) with fiber R2n. Define j on this bundle as follows: Any element in the latter can be N P written as a finite sum of reducible elements, 1kn zk ek, where zk 2 R2 and ek 2 E. This understood, define an almost complex structure on this tensor N N P P product bundle so as to send 1kn zk ek to 1kn (j0zk) ek, where   0 1 : j0 ¼ 1 0 The resulting complex bundle is EC. As advertised, EC is a rank n complex bundle. The C-action on EC can be seen by writing R2 as C so as to view EC as N N P (M  C) R E. Write a typical element as v ¼ k zk ek but now view zk 2 C. N P Then element s 2 C acts to send this element to sv ¼ k szk ek. Suppose that E is a complex bundle already. This is to say that it can be viewed as a real bundle with an almost complex structure j. Then E sits inside its complexification, EC, as a complex, rank n subbundle. In fact, EC has a L  is defined from the direct sum decomposition, linear over C, as E E, where E

6.4

Complex bundles over surfaces in R3

 is called the same real bundle as was E, but with j replaced by j. The bundle E complex conjugate bundle to E. To see E inside EC, observe that the endomorphism j acts as an endomorN N P P phism of EC sending any given v ¼ k zk ek to jv ¼ k zk jek. As an endomorphism now of EC, this j obeys j2v ¼ v, and it also commutes with multiplication by elements in C. This understood, the bundle E sits inside the bundle EC as the set of vectors v such that jv ¼ iv. Meanwhile,  sits in EC as the set of vectors v such that jv ¼ iv. the conjugate bundle E The inclusion of E into EC as a direct summand sends any given vector e 2 E to the N N vector 12 (1 e  i e).  and E are not isomorphic as complex bundles even though In general, E  R) are identical. their underlying real bundles (i.e., ER and E

6.4

Complex bundles over surfaces in R3

Introduce the Pauli matrices    0 i 0 t1 ¼ ; t2 ¼ i 0 1

  1 i ; t3 ¼ 0 0

 0 : i

These generate the quaternion algebra as t21 ¼ t22 ¼ t33 ¼ i ; t1 t2 ¼ t2 t1 ¼ t3 ; t2 t3 ¼ t3 t2 ¼ t1 ; t3 t1 ¼ t1 t3 ¼ t2 : Here, i denotes the 2  2 identity matrix. Let   R3 denote a given embedded or immersed surface. Assign to each point p 2  its normal vector P n(p) ¼ (n1, n2, n3). This assignment is used to define the map n: ! M(2; C) 2 given by n ¼ n1t1 þ n2t2 þ n3t3. Note that n ¼ i. Define a complex vector subbundle E    C2 to be the set of points (p, v) such that n(p)v ¼ iv. As explained momentarily, this is a complex vector bundle of rank 1 over . In general, if M is a smooth manifold and f ¼ (f1, f2, f3): M ! R3 with image in S2 is a smooth map, then there is the corresponding subbundle E  M  C2 P given by the set {(p, v) : f(p)v ¼ iv} where f ¼ jfjtj. What follows uses Definition 6.2 to verify that E does indeed define a complex vector bundle. Use for p the restriction of the projection from M  C2 to M. Likewise, use for m the restriction of the multiplication on C2 by elements in C. Take oˆ to be the map x ! (x, 0) 2 M  C2. To see about the final item, fix p 2 M and fix a unit length vector vp 2 Ejp  C2. This done, write the two components of vp as (ap, bp) and write those of any other vector v as (a, b). Now define the C-linear map Zp: C2 ! C by the rule that sends (a, b) 2 C2 to the complex number

63

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a¯pa þ b¯pb. As Zp restricts to Ejp as an isomorphism (over C), so it restricts to a neighborhood U  M of p so as to map each fiber of EjU isomorphically to C. Define lU to be this restriction. Here is another example of a map f from a compact surface to in R3 to S2: Let  denote the surface in question. Fix a point, x 2 R3, that lies in the interior of the region bounded by . A map f:  ! S2 is defined by taking for f(p) the unit vector (p  x)/jp  xj. Alternately, one can take x to lie outside of  to define a different version of E.

6.5

The tangent bundle to a surface in R3

The tangent bundle to a surface in R3 has an almost complex structure that is defined as follows: Let   R3 denote the surface and let n ¼ (n1, n2, n3) again denote its normal bundle. As explained in Chapters 3.4 and 3.5, the tangent bundle T can be identified as the set of pair (p, v) 2   R3 such that the dot product v n ¼ 0. Define j: T ! T by the rule j(p, v) ¼ (p, n  v), where  is the cross product on R3. Since n  n  v ¼ v, it follows that j is an almost complex structure. As a consequence, we may view T as a complex vector bundle of rank 1 over . This complex bundle is denoted often by T1,0.

6.6

Bundles over 4-dimensional submanifolds in R5

Introduce a set of five matrices in M(4; C) as follows: Written in 2  2 block diagonal form, they are:       0 tj 0 i i 0 ; g4 ¼ ; g5 ¼ i : For j ¼ 1; 2; 3: gj ¼ tj 0 i 0 0 i Here, {ta}1a3 again denote the Pauli matrices. Note that for each j, one has gj2¼ i. Note also that gigj þ gjgi ¼ 0 for i 6¼ j. Let M  R5 denote a dimension 4 submanifold; the sphere S4 is an example. Let n ¼ (n1, . . . , n5) denote the normal vector to M at any given point. Define the map n: M ! M(4, C) by P sending p to n(p) ¼ j njgj. Then n2 ¼ i. As a consequence, at each point in M, the matrix n has a 2-dimensional (over C) space of eigenvectors with eigenvalue þi. Define the bundle E  M  C4 to be the set {(p, v) : n(p)v ¼ iv.} This is a vector bundle over M with fiber C2.

6.8

6.7

Complex Grassmannians

Complex bundles over 4-dimensional manifolds

What follows uses the cocycle definition of a complex vector bundle to construct some rank 2 complex bundles over a given 4-dimensional manifold. Let M denote a 4-dimensional manifold. Fix a set finite set   M of distinct points. Assign to each p 2  an integer m(p) 2 Z. Also assign to each p 2 M a coordinate chart Up  M that contains p with diffeomorphism jp : Up ! R4 that sends p to 0. Make these assignments so that the charts for distinct pairs p and p0 are disjoint. To define a vector bundle from this data, introduce U to denote the open cover of M given by the sets U0 ¼ M and the collection {Up}p2. As the only nonempty intersections are between U0 and the sets from {Up}p2, a vector bundle over M with fiber C2 is defined by specifying a set of maps {g0p : U0 Up ! Gl(2; C)}p2. In order to specify these maps, first define a map g: C2 ! M(2; C) by the rule that sends   z1  z2 : z ¼ ðz1 ; z2 Þ ! gðzÞ ¼ z2 z1 ¨

Note that det(g(z)) ¼ jzj2 so g maps C2{0} to Gl(2; C). Use j0 to identify R4 with C2 so as to view g as a map from R4 to M(2; C) that sends R4{0} to Gl(2; C). Now let p 2 . Since jp(U0 Up) ¼ R4{0}, the composition x ! g(jp(x))m(p) for x 2 U0 Up defines a map from U0 Up to Gl(2; C). The latter map is defined to be g0p. ¨

¨

¨

6.8

Complex Grassmannians

Important examples of complex vector bundles involve the Cn analogs of the Grassmannians that are described in Chapter 1.6. To say more, fix integers m > 1 and n 2 {1, . . . , m1}; then set GrC(m; n) to denote the set of m-dimensional complex subvector spaces in Cn. As is explained momentarily, GrC(m; n) is a smooth manifold of dimension 2n(mn). Moreover, GrC(m; n) has over it a very natural, rank n, complex vector bundle. The latter sits in GrC(m; n)  Cm as the set E ¼ fðV; VÞ : V is an n-dimensional subspace of Cm and V 2 Vg: The bundle E is the C-analog of what is described in Chapter 3.3. This GrC(m; n) is called the Grassmannian of n-planes in Cm, and the bundle E is called the tautological n-plane bundle.

65

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The argument to prove that GrC(m; n) has the structure of a smooth manifold is much the same as the argument used in Chapter 1.6 to prove that the R versions of the Grassmannians are smooth manifolds. Likewise, the argument that E is a complex vector bundle over GrC(m; n) is much the same as that given in Chapter 3.3 for the R analog of E. Even so, it is worth summarizing both arguments because of a point that is referred to in a later chapter. What follows directly is a brief interlude to set some notation. To start, suppose that v and w are vectors in Cm. Their Hermitian inner product is denoted by v¯ w; it is defined using their respective components by  v w ¼  v1 w1 þ þ vm wm : Thus, the norm of a vector v 2 Cn is jvj ¼ (v¯ v)1/2. A basis (e1, . . . , en) for an n-dimensional subspace V  Cm is orthonormal when e¯i.ej ¼ 1 or 0 depending on whether i ¼ j or i 6¼ j. The orthogonal projection V : Cm ! V can be defined as follows: Fix an orthonormal basis, (e1, . . . , en) for V. Then X V ðwÞ ¼ ðej wÞ ej : 1jn To continue setting notation, introduce MC(m; n) to denote the space of C-linear homomorphisms from Cn to Cmn. This is to say that each point in MC(m; n) is a matrix with n columns and mn rows whose entries are complex numbers. The entries serve as coordinates and so identify MC(m; n) with Cn(mn). With this notation set, what follows next is a description of the smooth manifold structure on GrC(m; n). To start, define a topology on GrC(m; n) as follows: A basis of open neighborhoods for any given n-dimensional subspace V is labeled by the positive numbers, with the neighborhood labeled by a given e > 0 consists of the set of subspace V0 such that jV(v0 )  vj < e. A coordinate chart for a neighborhood of a given V  GrC(m; n) is the set O V of n-planes V0 such that V maps V0 isomorphically to V. A coordinate map, jV, from O V to R2n(mn) ¼ Cn(mn) ¼ MC(m; n) is defined as follows: Fix an orthonormal basis {ek}1kn for V so as to identify this vector space with Cn. Let V⊥  Cm denote the Hermitian conjugate space, thus the kernel of V. Fix an orthonormal basis {ua}1amn for V⊥ to make a C-linear identification of the latter with Cmn. Now let p denote a given point in Cn(mn). Viewed as a matrix in MC(m; n), it has components {pia}1in,1anm. The map j1 V sends p to the P n-dimensional subspace spanned by the set {ei þ 1an piaua}1in. Let V and V0 denote any two points in GrC(m; n). Fix orthonormal bases 0 {ei}1in and {ua0 }1amn for V0 and V0 ⊥ to define the coordinate chart map jV0 : O V0 ! MC(m; n) ¼ Cn(mn). This done, then the corresponding transition function jV0  j1 sends a given point p 2 jV(O V O V0 ) to the point V 0 p 2 jV0 (O V O V0 ) with the latter determined by p using the rule ¨

¨

66

6.8

p0ia ¼

X 1kn

Complex Grassmannians

  X 0  0a ek þ  ðG1 Þik u p u u b kb a 1bmn

where the notation is as follows: First, any given i, k 2 {1, . . . , n} version of Gik is given by X p e0 u : Gik ¼ e 0i ek þ 1an ia k a Second, G is the n  n matrix with (i, k) entry equal to Gik. Note that G  Gl(n; C) because j1 O V0. V (p) 2 O V The simplest example is that of GrC(m; 1), the set of 1-dimensional complex subspaces in Cm. This manifold is usually called the (m1)-dimensional complex projective space; and it is denoted by CPm1. This manifold has an open cover given by the collection of sets {O 1, . . . , O m} where O k is the version of what is denoted above by O V with V the complex line spanned by the vector in Cm with k’th entry 1 and all other entries 0. The complex span of a vector v 2 Cm is in O k if and only if the k’th entry of v is nonzero. Noting that MC(m; 1) ¼ Cm1, the O k version of the coordinate map j1 V sends (p1, . . . , pm1) 2 Cm1 to the line spanned by the vector (p1, . . . , pk1, 1, pk, . . . , pm1). Thus, the i’th entry of this vector is equal to pi when i < k, it is equal to 1 when i ¼ k, and it is equal to pi1 when k < i. For k0 < k, the matrix G for the intersection of O k with O k0 is a 1–1 matrix, this the nonzero, C-valued function on the set O k O k0 given by pk0 2 C{0}. The transition function 0 0 writes (p1, . . . , pm1) as the function of (p1, . . . , pm1) given as follows: ¨

¨

   

If i < k0 then pi0 ¼ pi / pk0 . If k0  i < k, then pi0 ¼ piþ1/pk0 . pk0 ¼ 1/pk0 . If k < i, then pi0 ¼ pi / pk0 .

What follows next is a brief description of the tautological bundle E ! GrC(m; n). The data needed for Definition 6.2 are a projection map to GrC(m; n), a zero section, a multiplication by C and a suitable map defined over a neighborhood of any given point from E to Cn. The bundle projection map p: E ! GrC(m; n) sends any given pair (V, v) with v 2 V and V an n-dimensional subspace of Cm to V. The zero section oˆ sends this same V 2 GrC(m; n) to the pair (V, 0). Multiplication by C on E by any given c 2 C sends (V, v) to (V, cv). For the final requirement, fix the basis {ei}1in for V as above. Let W 2 O V. Then the map lO V sends (W, w) 2 EjW to the vector in Cn with j’th entry equal to e¯j w. The map lO V just defined gives a C-linear map from EjO V to O V  Cn that respects the vector bundle structure of each fiber. These maps are used next to give a cocycle definition of E in the manner of Definition 6.3. A suitable chart for this consists of a finite collection of sets of the form O V for V 2 GrC(m; n).

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To define the transition functions, choose for each point V represented a basis, {ei}1in, for V. This done, suppose that V and V0 are two points in GrC(m; n) that are represented. The corresponding bundle transition function: gO V , O V : O V O V0 ! Gl(n; C) is the map G with entries {Gij}1i,jn as depicted above. 0

6.9

0

¨

68

The exterior product construction

The notion of the exterior product for Cn can be used to define a complex vector bundle of rank 2n over any given 2n-dimensional manifold. L To set the stage, let ∧*Cn ¼ k¼0,1, . . . ,n (∧kCn). Here, ∧0Cn is defined to be C. Meanwhile, ∧1Cn ¼ Cn and any given k > 1 version of ∧kCn is the k’th exterior power of Cn. Fix a basis {e1, . . . , en} for Cn and the vector space ∧kCn has a corresponding basis whose elements are denoted by fei1 ∧ ∧eik g1i1 < ::: 0 versions of ∧kCn by its action on the basis vectors given above. To define the latter, write z ¼ z1e1 þ þ znen. Then X z∧ei1 ∧ ∧eik ¼ z ðe ∧ðei1 ∧ ∧eik ÞÞ 1in i i

where ei ∧ (ei1 ∧ . . . ∧ eik) is zero if some ik ¼ i, and it is 1 times the basis vector obtained by adjoining i to the set {i1, . . . , ik} otherwise. The coefficient here is þ1 if the ordering (i, i1, . . . , ik) requires an even number of interchanges to make an increasing sequence, and 1 otherwise. Note that this 1 business guarantees that z ∧ (z ∧ ( )) ¼ 0. The second action of z is denoted in what follows by iz. This homomorphism is zero on ∧0Cn and maps any k > 0 version of ∧kCn to ∧k1Cn. It is defined ¯¯ W 2 ∧0Cn ¼ C. Now suppose that iz is inductively as follows: First, izW ¼ Z k n defined on ∧ C for k  1. To define it on ∧kþ1Cn it is enough to specify the action on elements of the form w ∧ o with w 2 Cn and o 2 ∧kCn. This understood, the homomorphism iz sends w ∧ o to (z¯ w) o  w ∧ iz(o). Use I( ) and the exterior product to define the map g:Cn ! End(∧*Cn) ¼ M(2n; C) by the rule

6.10

Algebraic operations

gðzÞ ¼ iz þ z∧: Noting that g2 ¼ jzj2 i, it follows that g maps Cn{0} to Gl(2n; C)). To continue with the preliminaries, suppose now that M is a given 2n-dimensional manifold. Fix a finite set, , of distinct points in M. For each p 2 , fix a coordinate chart Up  M centered at p with its map jp : Up ! R2n. Choose these charts so that they are pair-wise disjoint. Use the almost complex structure j0 to identify R2n with Cn so as to view any given p 2  version of jp as a map to Cn. The data just given will now be used to give a cocycle definition of a complex vector bundle with fiber dimension 2n. Take for the open cover the sets {Up}p2 with the extra set U0 ¼ M. The only relevant bundle transition functions are those for the sets {U0 Up}p2. For a given point p 2 , take the transition function gU0,Up to be the map from U0 Up to Gl(2n; C) that sends any given point x to g(jp(x)). ¨

¨

6.10

Algebraic operations

All of the algebraic notions from Chapter 4, such as bundle isomorphisms, homomorphisms, subbundles, quotient bundles, tensor products, symmetric products, antisymmetric projects and direct sums, have their analogs for complex vector bundles. The definitions are identical to those in Chapter 4 save that all maps are now required to be C-linear. There are two points that warrant an additional remark with regards to Definition 6.1. The first involves the notion of a bundle homomorphism. In the context of Definition 6.1, a complex bundle homomorphism is a homomorphism between the underlying real bundles that intertwines the respective almost complex structures. The second point involves the notion of tensor product. To set the stage, suppose that E and E0 are complex vector bundles over M of rank n and n0 respectively. Their tensor product is a complex vector N ER0 bundle over M of rank nn0 . Note, however, that the tensor product ER 0 of their underlying real bundles has dimension 4nn which is twice that N N of (E E0 )R. The latter sits in ER E0R as the kernel of the homomorphism N N 0 0 0 j i  i j where j and j denote the respective almost complex structures on E and E0 while i and i0 denote the respective identity homomorphisms on E and E0 . A third comment is warranted with regards to the exterior product. Recall that if E ! M is a real bundle of fiber dimension n, then ∧nE is a vector bundle of fiber dimension 1 over M. In the case when E is a complex vector bundle, then ∧nE is a vector bundle with fiber C. The latter is often denoted by det(E). The reason for this appelation is as follows: Suppose that E is given by the

69

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cocycle definition from Chapter 6.1. This is to say that a locally finite open cover, U, of M is given with suitable bundle transition functions from overlapping sets to Gl(n; C) with n the fiber dimension of E. Then ∧nE is defined by using the same open cover and with the transition function given by the determinant of the corresponding transition function for E. Note in this regard that any transition function for ∧nE must map to Gl(1; C) ¼ C{0}.

6.11

Pull-back

Let M and N denote smooth manifolds and let c: M ! N denote a smooth map. Chapter 5.1 explains how c is used to construct a real vector bundle over M from a given real vector bundle over N. If p: E ! N is the given bundle over N, then the corresponding bundle over M is denoted by c*E. If E has an almost complex structure, j, then there is a unique almost complex structure for c*E which is intertwined with j by the map cÐ : c*E ! E. Indeed, with (c*E) viewed in M  E as the set of pairs (x, v) with c(x) ¼ p(v), this almost complex structure acts to send (x, v) to (x, jv). Granted the preceding observation, it follows that the pull-back of any given complex vector bundle p: E ! N can be viewed in a canonical way as a complex vector bundle over M; and having done so, the covering map cÐ : c*E ! E is C-linear on each fiber. P By way of an example, return to the milieu of Chapter 6.4. Let  R3 denote a given embedded surface. As explained in Chapter 6.4, the association to each P P point in of its normal vector can be used to define a map n: ! M(2; C) P 2 and the vector subbundle E   C given by the set of points (p, v) with n(p)v ¼ iv. With this as background, remark that the assignment to p of its normal P vector also defines a map from to S2. The latter, n( ), is called the Gauss map. Let E1 ! S2 denote the subbundle of the product bundle S2  C2 given by the P set of points ((x1, x2, x3), v) with ( 1i3xiti) v ¼ iv. The bundle E is the pullback n*E1. P By way of a second example, suppose that  R3 is a embedded surface. P P Chapter 6.5 defines the complex rank 1 bundle T1,0 ! . The latter is n*T1,0S2. The following proposition gives a third application of pull-back. But for some notation, its proof is the same as that of its real bundle analog in Chapter 5.2. Proposition 6.5 Let M denote a smooth manifold, let n denote a positive integer, and let p: E ! M denote a given rank n, complex vector bundle. If m is sufficiently large, there exists a map cm: M ! GrC(m; n) and an isomorphism between E and the pull-back via cm of the tautological bundle over GrC(m; n).

Additional reading

Additional reading  Differential Geometry of Complex Vector Bundles, Shoshichi Kobayashi, Princeton University Press, 1987.  Characteristic Classes, John Milnor and James D. Stasheff, Princeton University Press, 1974.  Basic Bundle Theory and K-Cohomology Invariants, Dale Husemoller, Michael Joachim, Branislaw Jurco and Marten Schottenloher, Springer, 2009.  Algebraic Geometry: A First Course, Joe Harris, Springer, 2010.

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7

Metrics on vector bundles

A metric on Rn is a positive definite symmetric bilinear form. This is to say that if g is a metric, then g(v, w) ¼ g(w, v) and g(v, v) > 0 if v 6¼ 0. A metric on a real vector bundle E ! M is a section, g, of Sym2(E*) that restricts to each fiber as symmetric, positive definite quadratic form. Metrics always exist. To see this, let m ¼ dim(M) and let {ja : Ua ! Rm} denote a locally finite cover of M by coordinate charts such that {j1 a {x : jxj < ½}} is also a cover of M and such that EjUa over Ua has an isomorphism, ua : EjUa ! Ua  Rn. Use the function w from Appendix 1.2 to define wa : M ! [0, 1) so as to be zero on MUa and equal to w(jjaj) on Ua. Let d denote a given metric on Rn, and view d as a metric on the product bundle Ua  Rn for each a. A metric, g, on P E is g(v, w) ¼ a wa d(uav, uaw). Note in this regard that if g and g0 are metrics on E, then so is ug þ u0 g0 for any functions u and v with both u and u0 nonnegative functions such that u þ u0 > 0 at each point. A Hermitian metric on Cn is a bilinear form, g, on Cn with the following properties: First, g(u, cu0 ) ¼ c g(u, u0 ) and g(cu, u0 ) ¼ c¯g(u, u0 ) for all u, u0 2 Cn and all c 2 C. Second, g(u, u) > 0 for all u 2 Cn 6¼ 0. Finally, gðu;u0 Þ ¼ gðu0 ; uÞ for all u, u0 2 Cn. A Hermitian metric on a complex vector bundle p: E ! M is N a section of E* E* that restricts to each fiber as a Hermitian metric. Here is an equivalent definition: Let ER denote the underlying real bundle and let j denote its almost complex structure. A Hermitian on E is defined by a metric, gR, on ER if gR( , j( )) ¼ gR(j( ),( )). This condition asserts that j must define a skew-symmetric endomorphism on each fiber of ER. To obtain g from gR, N view E  ER by writing ER R C and then identifying E as the set of vectors of the form u ¼ e  i(je) with e 2 ER. Then inner product g(u, u0 ) is defined to be gðu; u0 Þ ¼ gR ðe; e0 Þ  igR ðe; je0 Þ: Note that if c ¼ c1 þ ic2 is any given element in C, then cu ¼ (c1e þ c2je)  ij(c1u þ c2ju). As a consequence, the preceding formula finds gðcu; u0 Þ ¼ c1 gR ðe; e0 Þ þ c2 gðje; e0 Þ  ic1 gR ðe; je0 Þ  ic2 gR ðje; je0 Þ;

7.1

Metrics and transition functions for real vector bundles

which is equal to c¯g(u, u0 ) by virtue of the fact that j is skew-symmetric. A similar computation finds that g(u, cu0 ) ¼ c g(u, u0 ) and also gðu;u0 Þ ¼ gðu0 ; uÞ. Meanwhile, a given Hermitian metric, g, on E defines the metric gR on ER by setting gR(e, e0 ) to be the real part of g(u, u0 ). With only notational changes, the construction given above of a metric on a real bundle will give a Hermitian metric on a complex bundle. If g and g0 are Hermitian metrics on a complex vector bundle, then so is u g þ u0 g0 when u and u0 are functions with u þ u0 > 0.

7.1

Metrics and transition functions for real vector bundles

A real bundle E ! M of fiber dimension n with a metric can be used to construct a locally finite, open cover U of M with two salient features: First, each U 2 U comes with an isomorphism from EjU to U  Rn. Second, the corresponding bundle transition functions map nonempty intersections of sets from U into O(n). To see why this is true, fix a locally finite cover, U, for E such that each U 2 U comes with an isomorphism from jU : EjU ! U  Rn. Use jU1 to define a basis of sections of EjU over U, and then use the Gram–Schmid procedure to construct a basis of sections of EjU over U which is orthonormal at each point as defined by the fiber metric. Let {sUj}1jn denote these sections. Define a new isomorphism EjU to U  Rn by declaring its inverse to be the map P that sends any given point (x, (v1, . . . , vn)) 2 U  Rn to jvjsUjjx. The corresponding bundle transition functions map any orthonormal vector in Rn to an orthonormal vector. As a consequence, all transition functions map to O(n). The question arises as to whether a trivializing chart can be found whose corresponding transition functions map to SO(n). Definition 7.1 A real vector bundle E ! M is said to be orientable if it has a trivializing cover such that the corresponding vector bundle transition functions on the overlaps have positive determinant. If a vector bundle is orientable, and if it has a fiber metric, then its transition functions can be chosen to map to SO(n). The following lemma says something about when a bundle is orientable: Lemma 7.2 A real vector bundle E ! M with some given fiber dimension n  1 is orientable if and only if the real line bundle det(E) ¼ ∧nE is isomorphic to the product bundle M  R. Proof of Lemma 7.2 Let U denote a locally finite cover of M such that each chart U 2 U comes with an isomorphism jU : EjU ! U  Rn. As noted above,

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these isomorphisms can be chosen so that the corresponding bundle transition functions map to O(n). The isomorphism jU induces an isomorphism cU : det(E)jU ! U  R. If U and U0 are two intersecting sets from U, then the corresponding vector bundle transition function for E is a map gU0 ,U : U \ U0 ! O(n). The corresponding bundle transition function for det(E) is the map from, U \ U0 to {1} given by det(gU0 ,U). This understood, if E is orientable, one can take U and the corresponding isomorphisms {jU}U2U such that det(gU0 ,U) ¼ 1 for all pairs U, U0 2 U. As a consequence, an isomorphism c: det(E) ! M  R is defined by declaring c on det(E)jU for any given U 2 U to be cU. To prove the converse, suppose that Z: det(E) ! M  R is an isomorphism. If U 2 U, then Z  cU1 can be written as (x, r) ! (x, ZUr) where ZU 2 {1}. The latter is such that ZU0 detðgU 0 ; U ÞZ1 U ¼1 on U \ U0 when U and U0 are any two sets from U with nonempty overlap. This understood, define a new isomorphism, j0 U : EjU ! U  Rn by j0 U ¼ sU  jU where sU at any given point in U is a diagonal matrix with entries (ZU, 1, . . . , 1) on the diagonal. It then follows from the preceding equation that the 0 }U2U map to SO(n). corresponding transition functions for the data {jU What follows are some examples.

7.1.1

The Mobius bundle

This bundle over S1 is defined in Chapter 1.3. It is not orientable. To see why, suppose to the contrary. This bundle would admit a section that has no intersections with the zero section. By definition, a section is a map v: S1 ! R2 whose coordinate entries (v1, v2) obey the condition cos(y)v1 þ sin(y)v2 ¼ v1 and sin(y)v1  cos(y)v2 ¼ v2 at each y 2 S1. This requires that v1 and v2 can be written as v1 ¼ rðyÞ cosð12 yÞ and v2 ¼ rðyÞ sinð12 yÞ with r: S1 ! R{0}. Such a pair is not continuous on the whole of S1.

7.1.2

The tautological bundle over RPn

This bundle is also not orientable. To see why, consider the map from S1 into RPn that sends the angle coordinate y 2 R / (2pZ) to the line with tangent ðcosð12 yÞ; sinð12 yÞ; 0; : : : ; 0Þ 2 Rnþ1 . The pull-back via this map of the tautological bundle is the Mobius bundle. Given that the Mobius bundle is not orientable, it follows that neither is the tautological bundle over RPn.

7.3

7.1.3

Metrics, algebra and maps

A bundle with an almost complex structure

As is explained next, a bundle with an almost complex structure is orientable. Indeed, let p: E ! M denote such a bundle and let 2n denote its fiber dimension. Recall from Chapter 6.2 that there is a locally finite cover U for M such that each U 2 U comes with an isomorphism to the product bundle, and such that all transition functions map to the group Gj  Gl(2n; R). As noted at the end of Chapter 2.4, matrices in Gj have positive determinant.

7.2

Metrics and transition functions for complex vector bundles

Let p: E ! M denote now a complex vector bundle with fiber dimension n (over C). Suppose that g is a Hermitian metric on E. Let U  M denote an open set and suppose that (s1, . . . , sn) is a basis for EjU is said to be orthonormal if g(si, sj) ¼ 1 when i ¼ j and zero otherwise. As the Gram–Schmid algorithm works just as well over C as over R, it follows that M has a locally finite cover such that E’s restriction to each set from the cover has an orthonormal basis. The following is a consequence: There is a locally finite cover, U, of M with two salient features: First, each U 2 U comes with a bundle isomorphism from EjU to U  Cn. Second, all transition functions map to the group U(n). Recall from Chapter 6.10 the definition of the complex bundle det(E) ¼ ∧nE. This bundle has fiber C. Lemma 7.3 The bundle det(E) is isomorphic to the product bundle E  C if and only if the following is true: There is a locally finite cover, U, of M such that each U 2 U comes with an isomorphism from EjU to U  Cn and such that all transition functions map to SU(n). The proof of this lemma is very much like that of Lemma 7.2 and so left to the reader.

7.3

Metrics, algebra and maps

Suppose that E is a vector bundle over a given manifold M with fiber Rn. Assume that E has a given fiber metric. The Nendows bundles the various L latter algebraically related bundles such as E*, m E, mE, ∧m E and Symm E with fiber metrics.

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Consider, for example, the case of E*. To obtain an orthonormal basis for the fiber of the latter at a given point in M, choose an orthonormal basis {e1, . . . , en} for E at the point in question. A corresponding basis for E* at the given point is given by the elements {u1, . . . , un} with the property that ui(ek) ¼ 0 if i 6¼ k and 1 if i ¼ k. Note that the metric on E at this point is given by the bilinear form N P ui. With these corresponding metrics on E and E*, the metric on E 1in ui can be viewed as an isometric isomorphism between E and E*. Consider for a second example the bundle ∧mE. To obtain an orthonormal basis for the fiber of the latter at a given point in M, fix an orthonormal basis {e1, . . . , en} for E at the point in question. The set {ei1 ∧ ∧ eim}1i1< 0 unless p ¼ q. To prove this last claim, go to a coordinate chart that contains p. Let (x1, . . . , xn) denote the coordinates in this chart, and view p as a point, z 2 Rn. Since q ¼ 6 p, there is a ball in Rn of some radius r > 0 that has z in its center, and does not contain the image of q via the coordinate chart. (This is certainly the case if q is not in the coordinate chart at all.) Then any path from p to q has image via the coordinate chart that enter this ball on its boundary and ends at the origin. The Euclidean distance of this image is no less than r. Meanwhile, because g is a metric, when written in the coordinates via the coordinate chart, g appears as a symmetric, bilinear form with P entries gij. As it is positive definite, there is a constant, c0, such that i,j gij(x)vivj 2 n  c1 0 jvj for any vector v 2 R and any point x in the radius r ball about z. This implies that the length of any path from q to p is at least c1/2 r > 0. 0 The function dist( , ) can be used to define a topology on M whereby the open neighborhoods of a given point p 2 M are generated by sets of the form {O p,e}e>0 where the set O p,e ¼ {q 2 M : dist(p, q) < e). The argument given in the P preceding paragraph (with the fact that i,j gij(x) vivj  c0 jvj2) implies that this metric topology on M is identical to the given topology.

8.2

Length minimizing curves

The following is the fundamental theorem that underlies much of the subject of Riemannian geometry. This theorem invokes two standard items of notation: First, repeated indices come with an implicit sum. For example, gijvivj means P i j ij 1 1ijn gijv v . Second, g denotes the components of the inverse, g , to a matrix g with components gij. Theorem 8.1 (the geodesic theorem) Suppose that M is a compact, manifold and that g is a Riemannian metric on M. Fix any points p and q in M.  There is a smooth curve from p to q whose length is the distance between p and q.  Any length minimizing curve is an embedded, 1-dimensional submanifold that can be reparametrized to have constant speed; thus g(g , g ) is constant.  A length minimizing, constant speed curve is characterized as follows: Let U  M denote an open set with a diffeomorphism j: U ! Rn. Introduce the Euclidean coordinates (x1, . . . , xn) for Rn and let gij denote the components of the metric in these coordinates. This is to say that the metric on U is given N by gjU ¼ j*(gij dxi dxj). Denote the coordinates of the j-image of a length

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minimizing curve by g ¼ (g1, . . . , gn). Then the latter curve in Rn obeys the equation: g¤ i þGjkm g k g m ¼ 0 where Gjkm ¼ 12 gjp ð@m gpk þ @k gpm  @p gkm Þ. Here, g i ¼ dtd gi and g¤ i ¼ d 2 gi . dt (Here, as above, repeated indices are implicitly summed.)  Any curve that obeys this equation in coordinate charts is locally length minimizing. This is to say that there exists c0 > 1 such that when p and q are two points on such a curve with dist(p, q)  c1 0 , then there is a segment of the curve with one endpoint p and the other q whose length is dist(p, q). 2

This theorem is called the geodesic theorem in what follows. A proof is given at the end of the next chapter. The next section of this chapter supplies an existence theorem for geodesics, and the remaining sections consider various examples where the geodesics can be described in very explicit terms. What follows directly are some additional remarks about the statement of the theorem. The equation depicted above in the theorem is called the geodesic equation. This equation refers to a particular set of coordinates. Even so, the notion that a map from an interval to M obeys this equation is coordinate independent. This is to say the following: Let U  M denote an open set with a diffeomorphism, j: U ! Rn. Now let I  R denote an interval and let gM : I ! U denote a smooth map such that g ¼ j  gM obeys the geodesic equation. Then so does the map g0 given by j0  gM if j0 : U ! Rn is any other diffeomorphism. You can use the Chain rule to prove this assertion if you make sure to account for the fact that the metric coefficients gij depend on the choice of coordinates. To say more about this, write the coordinates of c ¼ j  (j0 )1 as (c1, . . . , cn). Let gij and gij0 denote the components of the metric when written using the coordinates j and j0 . Then   k m   @c @c 0   g ðcð ÞÞ  : gij x0 ¼  0 @xi @xj km x

Meanwhile the respective components of g are given by (c1)i(g). Granted all of this, then the argument for coordinate independence amounts to a tedious exercise with the Chain rule. A geodesic is a map g, from an interval I  R into M that obeys the geodesic equation in one, and thus all coordinate charts that contain points in the image of g. Here is a last bit of commonly used terminology: The functions

1 jp j Gkm ¼ g ð@m gpk þ @k gpm  @p gkm Þ 2 1j;k;mn 0

that appear in the theorem’s statement are called the Christoffel symbols.

8.3

8.3

The existence of geodesics

The existence of geodesics

The proof given momentarily for the existence of geodesics invokes a fundamental theorem about differential equations. Theorem 8.2 (the vector field theorem) Let m denote a positive integer and let v: Rm ! Rm denote a given smooth map. Fix y0 2 Rm and there exists an interval I  R centered on 0, a ball B  Rm about the point y0, and a smooth map z: I  B ! Rm with the following property: If y 2 B, then the map t ! z(t) ¼ z(t, y) obeys d z ¼ vðzÞ: dt with the initial condition zjt¼0 ¼ y. Moreover, there is only one solution of this equation that equals y at t ¼ 0. The name vector field theorem refers to the fact that v can be viewed as a vector field on Rm. When viewed in this light, the image of the map t ! z(t) is said to be an integral curve of v. This vector field theorem is proved in Appendix 8.1. Proposition 8.3 below states a basic existence and uniqueness theorem for the geodesic equation. Proposition 8.3 Let M be a smooth manifold and let g denote a Riemannian metric on M. Let p 2 M and let v 2 TMjp. There exists e > 0 and a unique map from the interval (e, e) to M that obeys the geodesic equation, sends 0 to p, @ to v. and whose differential at 0 sends the vector @t Proof of Proposition 8.3 Fix a coordinate chart centered at p with coordinates (x1, . . . , xn) such that p is the origin. Let I  R denote an open interval centered at 0, and view the geodesic equation as an equation for a map z: I ! Rn  Rn that obeys a certain system of differential equations that involve only first derivatives. To obtain this system of equations, write the coordinates of z as (g, u). Then   d j ðg; uÞ ¼ u;  Gjk ðgÞuj uk : dt It is left as an exercise to verify that (g, u) obeys the latter equation if and only if g obeys the geodesic equation. This last equation has the schematic form m d dt z ¼ vðzÞ for a function z: I ! R where in this case m ¼ 2n. This understood, the assertions of Proposition 8.3 follow directly from the vector field theorem. The upcoming Proposition 8.4 states a stronger version of Proposition 8.3. As with Proposition 8.3, it is also a direct consequence of the preceding

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8 : Geodesics

theorem. To set the stage, introduce p: TM ! M to denote the vector bundle projection map. Suppose that O  TM is an open set. Theorem 8.1 above assigns the following data to each point v 2 O: First, a neighborhood of p(v), and second, a unique geodesic in this neighborhood that contains p(v) and whose tangent vector at p(v) is v. Thus, the points in O parametrize a family of geodesics in M. Among other things, the next proposition says that this family is smoothly parametrized by the points in O. Proposition 8.4 Let M be a smooth manifold and let g denote a Riemannian metric on M. Fix a point in TM. Then there exists an open neighborhood O  TM of this point, a positive number e > 0 and a smooth map gO: (e, e)  O ! M with the following property: If v 2 O, then the map gO( , v): (e,  e) ! M is a geodesic with gO(0, v) ¼ p(v) and with @  gO ð ; vÞ* @t  equal to the vector v. t¼0

Proof of Proposition 8.4 Fix a coordinate chart for a neighborhood of p(x) to identify the latter with a neighorhood of the origin in Rn and to identify TM on this neighborhood with Rn  Rn. As noted above, the geodesic equation can be viewed as an equation for a map, z ¼ (g, u), from an interval in R about the origin to a neighborhood of {0}  Rn in Rn  Rn that has the form   d j ðg; uÞ ¼ u; Gjk ðgÞuj uk : dt d z ¼ vðzÞ, an appeal to the theorem above proves As this equation has the form dt Proposition 8.4.

8.4

First examples

What follows gives some examples of metrics with their corresponding geodesics.

8.4.1

The standard metric on Rn

This is the Euclidean metric g ¼ dxi are the straight lines.

8.4.2

N

dxi. As you might expect, the geodesics

The round metric on Sn

The round metric is that induced by its embedding in Rnþ1 as the set {x 2 Rnþ1 : jxj ¼1}. This embedding identifies TSn as the set of pairs (x, v) 2 Rnþ1  Rnþ1

8.4

First examples

with jxj ¼ 1 and with x v ¼ 0. Here, x v ¼ 1jnþ1 xjvj. The round metric is such that the inner product between elements (x, v) and (x, w) in TSn is v w. Suppose g: R ! Sn  Rnþ1 is a smooth curve. The geodesic equation when written in coordinates g(t) ¼ (x1(t), . . . , xnþ1(t)) asserts that x¤ ´ j þ xj j x˙ j2 ¼ 0: To see that the solutions to this equation are the geodesics, view Rnþ1 as Rn  R and introduce the coordinate embedding y ! (y, (1  jyj2)1/2) of the ball of radius 1 in Rn into Sn  Rn  R. The round metric on Sn pulls back via this embedding as the metric on radius 1 ball in Rn with entries gij ¼ dij þ yiyj (1  jyj2)1/2. Here, dij ¼ 1 if i ¼ j and dij ¼ 0 otherwise. Thus, gij ¼ dij þ yiyj þ O(jyj4). As a consequence, Gijk ¼ djkyi þ O(jyj2). The equation asserted by the theorem is y¨ þ yj jy j2 þ O(jyj2) ¼ 0. This agrees with what is written above to leading order in y. Since the metric and the sphere are invariant under rotations of Sn, as is the equation for x above, this verifies the equation at all points. Note that this equation implies that t ! xj(t) lies in a plane. Indeed, if jx j ¼ 1 at time t ¼ 0, then this is true at all times t. Then the equation above finds that xj ðtÞ ¼ aj cosðtÞ þ bj sinðtÞ where jaj2 þ jbj2 ¼ 1 and ajbj ¼ 0. In particular, the geodesics are the curves in Sn that are obtained by intersecting Sn with a plane through the origin in Rnþ1.

8.4.3

The induced metric on a codimension 1 submanifold in Rnþ1

This last example where the geodesic equation for the sphere is written as an equation for a curve in Rnþ1 has an analog of sorts for any n-dimensional submanifold in Rnþ1. Let   Rnþ1 denote the submanifold in question. The relevant metric is defined as follows: Let x ! n(x) 2 Rnþ1 denote a smooth assignment to a point x 2  of a unit length unit normal vector to  at x. View T   Rnþ1  Rnþ1 as the set of pairs of the form (x, v) with x 2  and with v n(x) ¼ 0. The metric inner product between vectors (x, v) and (x, w) is declared to be v w. To say something about the geodesics, write a curve in  as a map from an interval in R to Rnþ1 as a map t ! (x1(t), . . . , xnþ1(t)) whose image lies in . The geodesic equation asks that the curve t ! x(t) in Rnþ1 obey x¤ j þ nj jx ð@ k ni Þjx x k x i ¼ 0: The proof that solutions to this equation give the geodesics in  is left to the reader. As a guide, remark that a proof can be had by choosing a very convenient coordinate chart about any given point, then writing this equation in the coordinate chart, and comparing it to the geodesic equation.

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8.4.4

The hyperbolic metric on Rn

P Let M  Rnþ1 denote the branch of the hyperbola where jxnþ1j2  1kn jxkj2 ¼ 1 and xnþ1 > 0. This manifold M is diffeomorphic to Rn with diffeomorphism given by the map that sends y 2 Rn to cðyÞ ¼ ðy; ð1 þ jyj2 ÞÞ 2 Rnþ1 ¼ Rn  R: In any event, view TM  TRnþ1 ¼ Rnþ1  Rnþ1 as the space of pairs (x, v) with x 2 M and vnþ1 xnþ1 

X 1kn

vk xk ¼ 0:

Granted this identification of TM, define a Riemannian metric on TM by the rule that makes the inner product between pairs (x, v) and (x, w) equal to X 1kn

vk wk  vnþ1 wnþ1 :

This metric is positive definite, as can be seen by writing P P 2 1/2 vnþ1 ¼ x1 nþ1 1knvkxk and using the fact that xnþ1 ¼ (1 þ 1knxk ) . I use gH in what follows to denote this metric. The metric gH just defined is called the hyperbolic metric. To justify this name, remark that the geodesics are as follows: Fix any point P 2 ¼ 1. and v 2 TMjx. Then (x, v) 2 TM  Rnþ1  Rnþ1 with 1knvk2  vnþ1 the curve t ! x(t) ¼ cosh(t)x þ sinh(t)v is a geodesic curve in M. Note that these curves are the intersections between M and the planes through the origin in Rnþ1. The proof that the curves just described are the geodesics can be had by using the coordinate chart given by the map c above. Note in this regard that it is enough to focus on the case where x ¼ (0, 1) 2 Rn  R by invoking symmetry arguments that are much like those used in the discussion above for the round metric on Sn. To say more about this, I need to introduce the Lorentz group, SO(n, 1). This is the subgroup of Gl(nþ1; R) that preserves the indefinite P quadratic form Z(x, x) ¼ jxnþ1j2  1kn jxkj2. This is to say that SO(n,1) consists of the matrices m 2 Gl(nþ1; R) for which Z(mx, mx) ¼ Z(x, x) for all x 2 Rnþ1. As it turns out, this is a Lie group whose dimension is 12 nðn þ 1Þ. In any event, mx 2 M if x 2 M and m 2 SO(n, 1). This is to say that the linear map mm : Rnþ1 ! Rnþ1 given by x ! mx restricts to define a diffeomorphism of M. Moreover, this diffeomorphism is such that m*m gH ¼ gH and so the length of a curve and the length of its mm image are the same. As a consequence, mm maps geodesics to geodesics. As explained momentarily, given x 2 M there exists m 2 SO(n, 1) such that mx is the point (0, 1) 2 Rn  R. Thus, one need only identify the geodesics through the latter point. What follows is a matrix m 2 SO(n, 1) that maps a given point x 2 M to the point (0, 1): Write x ¼ (x ^ , xnþ1) 2 Rnþ1 ¼ Rn  R. Let o 2 SO(n) denote a

8.5

Geodesics on SO(n)

matrix which is such that ox ^ ¼ (jx ^ j, 0, . . . , 0). Let o 2 SO(n, 1) denote the matrix that acts on any given v ¼ (v ^, vnþ1) 2 Rnþ1 ¼ Rn  R to give ov ¼ (ov ^, vnþ1). Now set p 2 SO(nþ1) to be the matrix that acts on (v1, v2, . . . , vnþ1) 2 Rnþ1 to give ^ jvnþ1 ; v2 ; :::; vn ; xnþ1 vnþ1 þ j x ^ jv1 Þ: ðxnþ1 v1  j x The matrix m ¼ po does the job.

8.5

Geodesics on SO(n)

To define the relevant metric, first define an inner product on the space M(n; R) of n n matrices by ha, a0 i ¼ tr(aTa0 ). This is a Euclidean metric. Fix an orthonormal basis. As I will be talking momentarily about SO(n) also, it is convenient to fix one so that each basis element is either a symmetric matrix or an antisymmetric one. (Note that the symmetric matrices are orthogonal to the antisymmetric ones.) Let {aj} denote the basis elements. Here, j runs from 1 to n2. Assume that the first n(n  1)/2 basis elements are antisymmetric. For each j 2 {1, . . . , n2}, let o j denote the 1-form on Gl(n; R) given by o j jm ¼ trðaj m1 dmÞ: As noted in Chapter 5.3, these forms are left-invariant 1-forms. Since the collection spans T*Gl(n; R) at the identity, so these forms span TGL(n; R) at any give point. Thus, they give an isomorphism of vector bundles TGl(n; R) ! Gl(n; R)  M(n; R) by the rule that associates to a pair P m 2 Gl(n; R) and v 2 TGl(n; R)jm the pair (m, i ai hoijm, vi) in the product Gl(n; R)  M(n; R). This basis {oj} gives the metric g ¼ ∑1in2 o

iN

oi

on Gl(n; R). Note that this metric does not depend on the chosen orthonormal basis since any one basis can be obtained from any other by the action of the group SO(n2) on the Euclidean space M(n; R). The metric g is invariant under the action on Gl(n; R) of left translation by elements in Gl(n; R). This means the following: If u 2 Gl(n; R), then the diffeomorphism lu : Gl(n; R) ! Gl(n; R) that sends m to um is such that l*g u ¼ g. i i Such is the case because l*o ¼ o . (In general, metrics on a manifold M are u sections of Sym2(T*M) and so pull-back under diffeomorphisms.) As noted in Chapter 5.5, the form o j restricts as zero on TSO(n)  TGL(n; R)jSO(n) when aj ¼ aTj . Meanwhile, the collection {oj : aTj ¼ aj}

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span TSO(n) along SO(n). This understood, introduce again A(n; R)  M(n; R) to denote the vector space of anti-symmetric matrices. The set {oj}1jn(n1)/2 restrict to SO(n) to give a bundle isomorphism TSO(n) ! SO(n)  A(n; R). It follows as a consequence that the restriction to TSO(n) TGl(n; R)jSO(n) of the metric defined above on TGl(n; R) defines the metric g¼

X 1inðn1Þ=2

o

iN

oi

on TSO(n). This metric is left invariant with respect to the left multiplication map on SO(n) because it is left invariant on Gl(n; R) and, in addition, because left multiplication on Gl(n; R) by an element in SO(n) maps SO(n) to itself. N P As an aside, note that this metric g ¼ 1in(n1)/2 oi oi on TSO(n) is also invariant with respect to the right multiplication map on SO(n). This is to say that it pulls back to itself under the right multiplication map ru on SO(n) that sends m ! mu1 with u any given element in SO(n). To see why this is so, remark first that i ¼ trðu1 ai u m1 dmÞ: r*o u

Note next that the map a ! u1au maps A(n; R) to itself when u 2 SO(n). Moreover, tr((u1au)T(u1a0 u)) ¼ tr(aTa0 ) for any a 2 M(n; R). Thus, the map a ! u1au acts as an isometry on A(n; R) when u 2 SO(n). This understood, it follows that u1 ai u ¼

X Sj a 1jnðn1Þ=2 i j

j

when ai 2 A(n; R) where S i are the entries of a (n(n1)/2)(n(n1)/2) orthogonal matrix, S. This implies that r*g u ¼

X X i jk

j

jN

Si Ski o

ok

where the sums range from 1 to n(n1)/2. Since S is orthogonal, the sum on the j right over i of SiSik is djk where djk here denotes the (n(n1)/2)  N (n(n1)/2) P j identity matrix. This is to say that r*g ¼ d o ok ¼ g. u 1j,kn(n1)/2 jk The metric g is said to be bi-invariant because it is both left and right invariant. As it turns out, any bi-invariant metric on SO(n) has the form cg with c > 0 a constant. Thus, the bi-invariant metrics are rather special. Return now to the issue of the geodesics for this metric g for TSO(n). The proposition that follows describes these geodesics. The statement of the theorem refers to the exponential map from M(n; R) to Gl(n; R) that sends a given matrix a 2 M(n; R) to ea ¼ iþaþ 12 a2 þ 3!1 a3 þ . As noted in Chapter 5.5, this map sends A(n; R) into SO(n) and it restricts to some small radius ball about the origin in A(n; R) as a diffeomorphism onto a neighborhood in SO(n) of i.

8.5

Geodesics on SO(n)

Proposition 8.5 Identify TSO(n) with SO(n)  A(n; R) as just described. With the metric g as given above, any solution to the geodesic equation on SO(n) and has the form t ! m eta where m 2 SO(n) and a 2 A(n; R). Conversely, any such map from R into SO(n) parametrizes a geodesic. The proof of this proposition (and of others to come) make use of an important observation about geodesics for induced metrics. To set the stage, suppose that X is a given Riemannian manifold and that M is a submanifold of X. Give M the induced metric. A geodesic in M need not be a geodesic in X. Witness, for example, the case of the round sphere in Rn as described in the previous part of this chapter. By contrast, a geodesic in X that lies entirely in M is a priori also a geodesic for the induced metric on M. This follows from the fact that geodesics are locally length minimizing. With the preceding understood, the proposition is proved by demonstrating that any path of the form m ! meta for m 2 Gl(n; R) and a 2 A(n; R) is a priori a geodesic in Gl(n; R). Note that this last result implies the assertion made by the proposition for the following reason: If every path through a given matrix m 2 SO(n) of the form t ! meta with a 2 A(n; R) is a geodesic, then Proposition 8.3 guarantees that such paths account for all of the geodesics that pass through m. By way of notation, a submanifold M in a Riemannian manifold X is said to be totally geodesic when all geodesics in M as defined by the induced metric from X are also geodesics for the metric on X. The proof given here of the proposition demonstrates that SO(n) is a totally geodesic submanifold in Gl(n; R).

8.5.1

Geodesics on Gl(n; R)

The proposition that follows describes certain of the geodesics on Gl(n; R). Proposition 8.6 Identify TGl(n; R) with Gl(n; R)  M(n; R) as described above. With the metric g as above, paths of the form t ! m eta with m 2 Gl(n; R) and a 2 M(n; R) obeying a aT  aTa ¼ 0 are geodesics. In particular, paths of the form t ! m eta are geodesics in Gl(n; R) if a ¼ aT. By way of notation, if a and a0 are any two n  n matrices, the matrix aa0  a0 a is said to be their commutator and is written [a, a0 ]. Proof of Proposition 8.6 The metric is left invariant and this has the following consequence: If I is an interval containing 0 and g: I ! Gl(n; R) is a solution to the geodesic with g(0) ¼ m, then the map t ! Lm-1(g(t)) is a solution to the geodesic equation that hits the identity matrix i at t ¼ 0. This understood, no generality is lost by showing that a solution, g, to the geodesic equation with g(0) ¼ i has the form t ! eta if [a, aT] ¼ 0.

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To see this, remember that the map from M(n; R) ! Gl(n; R) that sends a to e gives a diffeomorphism of some neighborhood of i in Gl(n; R) with a ball about the origin in M(n; R). This understood, it is enough to prove that any line through the origin in the vector space M(n; R) of the form t ! ta with [a, aT] ¼ 0 is a geodesic for the metric that is obtained by pulling back the metric g on Gl(n; R). As a line obeys g¨ i ¼ 0, so it is sufficient to prove that the term Gijkg jg k is also zero when g is the line t ! t a with [a, aT] ¼ 0. To show this, consider first the pull-back of g via the map a ! ea. Recall from Chapter 5.5 that the pull-back of the 1-form oq for q 2 M(n; R) via the exponential map is the 1-form a

ð1

uq ¼ trðesa qesa daÞds: 0

It follows as a consequence that the metric g appears with respect to this coordinate chart as ð1 ð1 gja ¼

0

trðesa daT esa es a daes a Þdsds0 : T

T

0

0 0

This is to say that ð1 ð1 gij ja ¼

0

0

trðesa aTi esa es a aj es a Þdsds0 : T

T

0 0

Granted the preceding, what follows explains why Gijk g ig j is zero when t ! g(t) ¼ ta with a obeying [a, aT] ¼ 0. To show this, digress momentarily and fix elements q1, q2 2 M(n, R). Consider the function on M(n; R) given by 0

0

a ! trðesa qT1 esa es a q2 es a Þ : T

T

Note that the integrand above that defines the pull-back at a of the metric g involves this sort of expression. The directional derivative of the function just described in the direction of q 2 Gl(n; R) is given by ð1 T T T T T T 0 0 s dt trððeð1tÞsa qT etsa qT1 esa þ esa qT1 eð1tÞsa qT etsa Þes a q2 es a Þ 0

þs

0

ð1 0

0

0

0

0

0

0

dt trðesa qT1 esa ðeð1tÞs a qets a q2 es a  es a q2 eð1tÞs a qets a ÞÞ: T

T

8.6

Geodesics on U(n) and SU(n)

This last observation is relevant because the evaluation Gijkg jg k requires evaluating expressions of this sort when two of q1, q2 or q have the form P i i g ai. Note in particular, that this last sum is a when g is the line t ! ta. Granted the preceding consider the expression above first when q and either of q1, q2 are multiples of a. If this is the case, then the cyclic property of the trace implies that the integrand above is zero. Meanwhile, if q1 and q2 are multiples of a, then what is written above is proportional to ð1

0

0

dt trððseð1tÞsa qT eð1tÞsa þ s0 eð1tÞs a qeð1tÞs a Þ½a;aT Þ : T

T

0

In particular, the integral of this last expression with respect to s and s0 is zero since the result can be written as tr(b [a, aT]). Thus Gijk g j g k ¼ 0 for the line t ! ta when [a, aT] ¼ 0.

8.6

Geodesics on U(n) and SU(n)

The story for these groups is very much the C analog of what was done just now for Gl(n; R) and SO(n). To start, define a Hermitian inner product on the vector space M(n; C) of n n matrices with complex entries by the rule (a, a0 ) ! tr(a{a0 ). Fix an orthonormal basis {ai}1in2 for M(n; C) with respect to this Hermitian inner product. For each integer i 2 {1, . . . , n2}, let o j denote the C-valued 1-form on Gl(n; C) given by oi jm ¼ trðai m1 dmÞ: As noted in Chapter 5.6, each of these forms is a left invariant 1-form on Gl(n; C). As their real and imaginary parts span T*Gl(n; C)ji, so their real and imaginary parts span T*Gl(n; C) at each point. This understood, their real and imaginary parts define an isomorphism from TGl(n; C) to the product bundle Gl(n; C)  M(n; C). The isomorphism in question sends a given tangent vector P v at a given point m 2 Gl(n; C) to the pair (m, 1in2 ai hoi, vi). Meanwhile, the collection {o j} defines the metric oi g ¼ ∑1in2 reð

O

oi Þ

on TGl(n; C). Note that g does not depend on the chosen Hermitian basis because any such basis is obtained from any other by the action of some element in U(n2). As an aside, note that T M(n; C) has an almost complex structure such that the resulting complex, rank n2 vector bundle is spanned at each point by

89

90

8 : Geodesics @ the entries of the matrix @m . As an open set in M(n; C), the tangent bundle to Gl(n; C) inherits this almost complex structure. This understood, the isomorphism defined by the forms {oi} from TGl(n; C) to Gl(n; C)  M(n; C) is C-linear, and thus an isomorphism of complex vector bundles. The metric g just described is defined on the underlying real tangent bundle of Gl(n; C) N i P i  o on the complex bundle. by the Hermitian metric 1in2 o To say more about the metric g, introduce from Chapter 2.4 the almost complex structure j0 2 M(2n; R) so as to view M(n; C) as the subvector space Mj  M(n; 2R) of matrices that obey mj0  j0m ¼ 0. In particular, recall that m 2 Mj if and only if its entries obey m2k,2i ¼ m2k1,2i1 and m2k,2i1 ¼ m2k1,2i for any pair of i, k from {1, . . . , n}. This understood, the identification between M(n; C) and Mj associates to a given a 2 M(n; C) the matrix m 2 Mj with m2k1,2i1 and m2k,2i1 given by the respective real and imaginary parts of a. This identification sends Gl(n; C)  M(n; C) to the subspace Gj  Mj of invertible elements. In particular, it follows directly from all of this that the metric g defined above on Gl(n; C) is the restriction to Gj  Gl(2n; R) of the metric that was defined above in the preceding part of this chapter.

8.6.1

The geodesics on Gl(n; C)

The next proposition describes some of the geodesics for the metric just described. Proposition 8.7 Identify TGl(n; C) with Gl(n; C)  M(n; C) as just described. With the metric g as above, any path of the form t ! m eta with m 2 Gl(n; C) and a 2 M(n; C) obeying [a, a{] ¼ 0 is a geodesic. Proof of Proposition 8.7 There are two ways to argue. Here is the first: Suppose that a 2 Mj and m 2 Gj. The path t ! m eta in Gl(2n; R) lies entirely in Gj, and so if it is a geodesic in Gl(2n; R), then it is a geodesic in Gj as defined using the metric on the latter that is induced by the inclusion Gj  Gl(2n; R). It is a geodesic in Gl(2n; R), if it is the case that [a, aT] ¼ 0. This understood, let aC denote a’s counterpart in M(n; C). As explained momentarily, the commutator { [a, aT] ¼ 0 in Gl(2n; R) if and only if [aC, aC] ¼ 0 in Gl(n; C). As noted in 0 . Chapter 2.4, the product bb0 of matrices in Mj is in Mj and obeys (bb0 )C ¼ bCbC ta taC Thus (me )C ¼ mCe , and so the paths in Gl(n; C) as described by the proposition are geodesics. { To see that [a, aT] ¼ 0 in Gl(2n; R) if and only if [aC, aC] ¼ 0 in Gl(n; C), note first that the matrix j0 as defined in Chapter 2.4 obeys j0T ¼ j0. This implies { that aT 2 Mj if a 2 Mj. It is left for the reader to verify that (aT)C ¼ aC . Granted { these last points, it then follows [a, aT] 2 Mj and ([a, aT])C ¼ [aC, aC].

8.6

Geodesics on U(n) and SU(n)

The second proof is the C analog of the argument for the Gl(n; R) version of this proposition. The only substantive difference is that the expression for the metric in the case of Gl(n; C) replaces each occurrence of the transpose of a matrix with the Hermitian conjugate of the matrix. To elaborate just a little, as with Gl(n; R), the fact that the metric is invariant with respect to the left translation diffeomorphisms implies that it is sufficient to prove that the paths t ! eta are geodesics if a 2 M(n; C) obeys [a, a{] ¼ 0. To see if this is the case, recall from Chapter 5.6 that the exponential map a ! ea from the vector space M(n; C) into Gl(n; C) restricts to a small radius ball about 0 in M(n; C) as a diffeomorphism from said ball onto a neighborhood of the identity matrix in Gl(n; C). The metric pulls back via this map to ð1 ð1 gja ¼ re

0

trðesa da{ esa es a daes a Þdsds0 : {

{

0

0 0

Note in particular that Hermitian adjoints appear in this formula, not transposes. But for this replacement here and subsequently the argument for the proposition is identical to that used to prove the Gl(n; R) version of the proposition. As such, the details are left to the reader.

8.6.2

The geodesics on U(n) and SU(n)

View U(n) as the subgroup in Gl(n; C) of matrices m with m1 ¼ m{ and view SU(n)  U(n) as the subgroup of matrices with determinant 1. As such, the metric on Gl(n; C) described above induces metrics on both U(n) and SU(n). Note in this regard that the 1-form tr(a m1dm) restricts to zero on TU(n) if a{ ¼ a, and it is nowhere zero and real valued on TU(n) if a 6¼ 0 and a{ ¼ a. Such a form is nonzero on TSU(n) if and only if a is not a multiple of the identity. Keeping this in mind, let {ai}1in2 denote an orthonormal basis for the vector space A(n; C)  M(n; C), this the vector subspace (over R) of skew Hermitian matrices. Thus a{ ¼ a when a 2 A(n; C). Agree to take ai to have trace zero when i < n2 and to take an2 ¼ pi ffi i. The induced metric on N N n P P U(n) is g ¼ 1in2 oi oi, and that on SU(n) is g ¼ 1in21 oi oi. Let m 2 U(n) and let a 2 A(n; C). As noted in Chapter 5.6, the matrix mea is in U(n). Moreover, if m 2 SU(n) and a has zero trace, then m ea is in SU(n). This is background for the proposition that follows. Proposition 8.8 The geodesics for the metric defined above on U(n) are of the form t ! m eta where m 2 U(n) and a 2 A(n; C). The geodesic for the metric just defined on SU(n) are of this form with m 2 SU(n) and a a traceless matrix in A(n; C).

91

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Proof of Proposition 8.8 Because the curves described are geodesics in Gl(n; C), they are a priori geodesics in the subgroup in question. The first proposition in Chapter 8.3 above guarantees that these account for all of the geodesics.

8.7

Geodesics and matrix groups

A Lie group G is said in what follows to be a compact matrix group if it appears as a subgroup of some n  1 version of either SO(n) or U(n). Thus, G is a submanifold with the property that m1 2 G if m is a matrix in G, and also mh 2 G when both m and h are in G. What was said above about the geodesics in SO(n), U(n) and SU(n) has an analog for any compact matrix group. The metric here is assumed to be the one induced by the metric on SO(n) or U(n) that is discussed in Chapters 8.5 and 8.6. This is to say that a tangent vector to G is a priori a tangent vector to the linear group, and so the metric used in Chapters 8.5 and 8.6 can be used to define its norm. To elaborate, remark first that TGji appears as a subvector space (over R) in the vector space of rank n  n real, antisymmetric matrices, or complex antiHermitian matrices, as the case may be. It is customary to use lie(G) to denote this vector space. The following proposition describes the geodesics in the general case: Proposition 8.9 Suppose that G is a compact matrix group, and let m 2 G.  The map a ! m ea from lie(G) to the general linear group has image in G.  This map restricts to some ball about the origin in lie(G) as a diffeomorphism of the latter onto a neighborhood of m in G.  For any given a 2 lie(G), the map t ! m eta from R to G is a geodesic; and all geodesics through m 2 G are of this sort. Proof of Proposition 8.9 The second and third bullets of Proposition 8.9 follow with the verification of the first bullet using the same arguments given above for the G ¼ SO(n) or U(n) or SU(n) cases. To see about the first bullet, note first that it is enough to consider the case when m is the identity. To prove the assertion in the latter case, it is enough to prove that the map a ! ea maps lie(G) into G. Indeed, as the differential of this map at the origin is the identity on lie(G), it then follows from the inverse function theorem (Theorem 1.1) that this map restricts to a ball about the origin in lie(G) as a diffeomorphism onto a neighborhood in G of i. Granted the preceding, fix some very large integer, N, and write ea as (ea/N)N. Given the definition of lie(G) as TGji, it follows that ea/N can be written as gN þ rN where gN 2 G and where jrNj  c0 N2. As a consequence, ea ¼ (gN)N þ wN

Appendix 8.1

The proof of the vector field theorem

where jwNj  c0N1. As each gN 2 G, so (gN)N 2 G. Taking N ! 1 shows that ea 2 G also.

Appendix 8.1

The proof of the vector field theorem

The primary purpose of this appendix is to prove the vector field theorem (Theorem 8.2) in Chapter 8.3. By way of reminder, this theorem concerns solutions to an equation for a map from a neighorhood of the origin in R to Rm of the following sort: Let v: Rm ! Rm denote a given smooth map, and let z denote a map from a neighborhood of the origin in R to Rm. The map z is asked to obey the equation d z ¼ vðzÞ: dt Proof of the vector field theorem (Theorem 8.2) Let B  Rn denote the 0 (I; Rm) denote the space radius 2 ball. Fix e > 0 and let I ¼ [e, e]. Let C * m m of continuous maps from I to R that equal 0 2 R at 0 2 I. Keep in mind that this is a complete metric space where the topology is defined by the distance function that assigns to maps u and w the distance ku  w k ¼ 0 0 supt2I ju(t)  w(t)j. Define a map, T: B  C * (I; Rm) ! C * (I; Rm) by the rule ð Tðy; uÞjt ¼ vðy þ uðsÞÞds : ½0;t

If T(y, u) ¼ u and if u is a smooth map, then the map t ! z(t) ¼ y þ u(t) obeys the desired equation with initial condition z(0) ¼ y. Conversely, a solution to the desired equation with z(0) ¼ y is a fixed point of the map u ! Ty(u) ¼ T(y; u). Existence of a fixed point: To see that there is a fixed point, let c0  1 denote a 0 constant such that jv(z)j  c0 when jzj  1. Let B1 denote the ball in C * (I; Rm) with radius 1 centered about the constant map. On this ball,     Ty ðuÞjt   jtj c0  e c0 : Thus Ty maps B1 to itself if e < c1 0 . To see that Ty is a contraction for small e, let c0  1 be such that jv(z)  v(z0 )j  c0 jz  z0 j when jzj and jz0 j are both less than 1. Then     ðTy ðuÞ  Ty ðu0 ÞÞjt   c0 t supjsjt juðsÞ  u0 ðsÞj  c0 t ku  u0 k: 0 0 Thus if e < c1 0 , then kTy(u)Ty(u )k  ku  u k and Ty is uniformly contracting. As a consequence, it has a unique fixed point on the ball B1.

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Fixed points are smooth maps: To see that a fixed point is smooth, note that what is written on the right-hand side (8.2) is a C1 function. So, if u is equal to what is written on the right-hand side of (8.2), then u is a C1 function. This implies that the right-hand side of (8.2) is a C2 function. But then u is a C2 function and so the right-hand side of (8.2) is a C3 function. Continuing in this vein shows that u is infinitely differentiable. It follows, in particular, that any fixed point of the map U ! Ty(U) obeys the desired equation. There is a unique solution with z(0) ¼ y: To prove this, suppose that t ! z(t) is a solution to the equation for t 2 I. I need to prove that u ¼ z  y is in B1, for then the result follows from the contraction mapping theorem (Theorem 1.6). To see that u is in B1, I need to prove that ju(t)j is small for t 2 I. To see that such is the case, let s 2 (0, 1) denote the smallest value of t such that ju(t)j ¼ 1. Since u is a fixed point of the map Ty, it follows that juðsÞj  c0 jsj and so if the constant e that defines the interval I is chosen so that e < c1 0 , then this shows that s > e and so u 2 B1. The dependence on y 2 B: Let y ! uy now denote the mapping that associates to each y 2 B the unique fixed point of the map Ty in B1. It is a consequence of U being a fixed point and the fact that jv(z)  v(z0 )j  c0 jz  z0 j when both jzj and jz0 j are less than 1 that kuy  uy0 k  c0 ðjy  y0 j þ e kuy  uy0 kÞ thus, given that e < c1 0 , this tells us that the assignment y ! uy is Lipschitz 0 as a map to the metric space C * (I, Rm). In particular, for each t 2 I, one has juy(t)  uy0 (t)j  c0 jy  y0 j. To say something about higher-order derivatives, one takes successively more involved difference quotients. This is straightforward but tedious, and I leave it to you.

Additional reading  Introduction to Smooth Manifolds, John M. Lee, Springer, 2002.  An Introduction to Riemannian Geometry and the Tensor Calculus, C. E. Weatherburn, Cambridge University Press, 2008.  Elementary Differential Geometry, A. N. Pressley, Springer, 2010.  Introduction to Differentiable Manifolds, Louis Auslander and Robert F. MacKenzie, Dover Publications, 2009.  Three Dimensional Geometry and Topology, William Thurston and Silvio Levy, Princeton University Press, 1997.

Additional reading

 The Shape of Space, Jeffrey R. Weeks, CRC Press, 2001.  Differential Geometry, Lie Groups and Symmetric Spaces, Sigurdur Helgason, American Mathematical Society, 2001.  A Course in Metric Geometry, Dmitri Burago, Yuri Burago and Sergei Ivanov, American Mathematical Society, 2001.

95

9

Properties of geodesics

This chapter discusses various notions that involve geodesics. The final parts of this chapter use some of these to prove the geodesic theorem (Theorem 8.1).

9.1

The maximal extension of a geodesic

Let I  R denote a closed interval that is not the whole of R, and let s denote an end point of I, for the sake of the discussion, the maximum point. Let g: I ! M denote a geodesic. It follows from Proposition 8.3 that there exists an extension of g to an interval with maximal endpoint s þ e for some e > 0. To see this, take a point q ¼ g(s0 ) for s0 nearly s. Then q is very close to p ¼ g(s). Proposition 8.3 tells us that the geodesic t ! gq(t) ¼ g(s0 þ t) is defined for jtj  e for some e that depends only on p. This extends g. It follows as a consequence that each geodesic in M has a maximum extension. In particular, if M is compact, then each geodesic extends as a geodesic mapping R into M. A manifold M is called geodesically complete if each geodesic can be extended so as to map the whole real line into M. All compact manifolds are geodesically complete. To see a manifold with metric that is not geodesically complete, take M to be the complement in Rn of any nonempty set but take the Euclidean metric from Rn. In general, the complement of any point in a geodesically complete manifold is not geodesically complete.

9.2

The exponential map

The following proposition holds when M is geodesically complete: Proposition 9.1 There exists a smooth map, e: R  TM ! M with the following property: Fix v 2 TM. Then the corresponding map ev ¼ e( , v): R ! M is the

9.2

The exponential map

unique solution   to the geodesic equation with the property that ev(0) ¼ p(v) @  and ðev Þ* @t ¼ 1.  t¼0

Proof of Proposition 9.1 The map e is well defined because M is geodesically complete. The issue is whether e is smooth. This is a local question in the following sense: The map e is smooth if and only if it is smooth near any given point in its domain. To see that such is the case, fix a point v 2 TM, a neighborhood O  TM of v and e > 0 such that e is smooth on (e, e)  O. Proposition 8.4 can be used to obtain e. Let T  e denote either 1 or the ˆ is smooth for any open smallest number such that e’s restriction to [T, T]  O ˆ set O  TM that contains v. The claim made by the proposition follows with a proof that T ¼ 1. To see that T ¼ 1, suppose to the contrary that T is finite so as to derive some @ nonsense. Let p ¼ eV(T) ¼ e(T, v) and let vT 2 TMjp denote the vector (ev)* @t at t ¼ T. Fix an open set OV  TMjp containing vT. According to Proposition 8.4, there exists d > 0 and a smooth map, gOV, from (d, d)  OV into M such that if v0 2 OV, then gOV( , v0 ) is a geodesic that starts at p(v0 ) 2 M and has initial vector v0. With the preceding understood, fix an open neighborhood O0  TM of v such that e is smooth on ðT  d; T  12 dÞ  O0 . Define the map eˆ to map this same @ domain into TM by the rule ^eðt; v0 Þ ¼ e* @t . Thus, p  eˆ ¼ e*. There is then a neighborhood O00  O0 such that ^e maps fT  34 dg  O0 0 into Ov. This understood, the uniqueness assertion in Proposition 8.3 implies that        3 3 1 when v00 2 O00 and s 2  d; d : e T þ s  d; v00 ¼ gOV s;^e T  d; v0 4 4 4 This exhibits eˆ on ð0; T þ 14 dÞ  O00 as the composition of two smooth maps, and hence smooth. This is nonsense given the definition of T. Given the map e, define the exponential map, exp : TM ! M by the rule exp(v) ¼ e(1, v). This is to say that exp(v) is the point in M that is obtained from p(v) by traveling for time t ¼ 1 from p(v) along the geodesic that starts at p(v) at time 0, has initial direction v at p(p) and has speed jvj. When p 2 M has been specified, the restriction of exp to TMjp is denoted in what follows by expp. The map exp is called the exponential map because this map is given by the exponential of a matrix when M is SO(n) or U(n) or one of their subgroups. This identification of maps follows from what is said about the geodesics on such groups in Chapters 8.5 to 8.7.

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9.3

Gaussian coordinates

The exponential map can be used to obtain a preferred coordinate chart for a neighborhood of any given point in M. This is because the exponential map restricts to a small radius ball in the fiber of TM over any given point as a diffeomorphism. To see that such is the case, fix a point p 2 M. Then it is enough to prove that the differential of expp at the v ¼ 0 in TMjp is an isomorphism. In fact, it is an isometry. To elaborate, remark that T(TMjp) is canonically isomorphic to p*(TMjp) where p: TM ! M is the projection map. This follows from the fact that the pull-back p*(TMjp) ! TMjp consists of pairs (v, v0 ) 2 TMjp. The differential at 0 2 expp sends v 2 TMjp to the push-forward of the vector @ @t at s ¼ 0 via the map s ! expp(sv). At any given s, the point expp(sv) 2 M is the point given by e(1, sv) with e as defined in the preceding section. This is the @ jt¼0 Þ ¼ v. same point as e(s, v). As a consequence, ((expp)* v)j02TMjp is ðeð ; vÞ* @t n Fix an isometric identification between TMjp and R . The image via a small radius ball in Rn via the exponential map gives a diffeomorphism between this ball and an open neigborhood of p in M. The inverse of this diffeomorphism gives what are called Gaussian or normal coordinates centered at p. You may see the term injectivity radius at p used with regards to Gaussian coordinates. This is the least upper bound on the radii of a ball in Rn about the origin which is mapped diffeomorphically onto its image by expp. As is demonstrated next, the metric gij in these coordinates and the geodesics have a particularly nice form. Proposition 9.2 Let (x1, . . . , xn) denote Gaussian coordinates centered at any given point in M. Then the metric in these coordinates has the form gij ¼ dij þ Kij(x) where Kij(x) xj ¼ 0 and also jKijj  c0jxj2. Moreover, the geodesics through the given point appear in these coordinates as the straight lines through the origin. Proof of Proposition 9.2 The fact that the straight lines through the origin are the geodesics through the given point follows directly from the definition of the exponential map. As noted above, the differential of the exponential map at the origin in TMjp is the identity, and this implies that gij ¼ dij þ Kij where jKj  c0jxj. To see that it is O(jxj2), suppose that Kij ¼ Kij,kxk þ O(jxj2). Note that Kij,k ¼ Kji,k. Then Gijk ¼ 12 ðKij;k þ Kik;j  Kjk;i Þ þ OðjxjÞ. This being the case, then the line t ! t vi with v a constant vector can be a geodesic only if Kij;k vj vk  12 Kjk;i vj vk ¼ 0. Taking all possible v, this says that Kij,k þ Kik,j  Kjk,i ¼ 0. A little linear algebra will show that this equation can be satisfied for all i, j and k only if Kij,k ¼ 0 for all i, j, k. Indeed, first take i ¼ j ¼ k shows that K11,1 ¼ 0, and so all cases where i ¼ j ¼ k are zero. Next take i ¼ 1

9.3

Gaussian coordinates

and j ¼ 1 and k ¼ 2 to see that K11,2 ¼ 0. With the latter zero, taking i ¼ 2 and j ¼ k ¼ 1 finds that K21,1 ¼ 0. By symmetry, K12,1 ¼ 0. Thus, all cases where two of i, j and k agree are zero. Now consider the case where where i, j, and k are distinct, say 1, 2, 3. Then i ¼ 1, j ¼ 2 and k ¼ 3 finds that K12,3 þ K13,2  K23,1 ¼ 0, while i ¼ 2, j ¼ 3 and k ¼ 1 finds K23,1 þ K21,3  K13,2 ¼ 0. Adding these equations and using the symmetry Kij,k ¼ Kji,k tells us that K21,3 ¼ 0. Thus, all cases where i, j and k are distinct are zero. n To see that Kij(x)xi ¼ 0, fix e > 0 but with e < c1 0 . Fix v 2 R with small norm; so that the point x ¼ v lies in the Gaussian coordinate chart. Let t ! Z(t) denote a smooth map from [0, 1) to Rn such that Z(0) ¼ 0 and Z ¼ 0 for t  e. For each s 2 (e, e) consider the path g[s] given by t ! tv þ sZ(t). If e is small, then this will be a path in the coordinate chart for all t with jtj  1. The s ¼ 0 version is a geodesic, and as a consequence, the function S ! ℓg½s ¼

ð1 

gij ðg½s Þ g ½s i g ½s j

1=2

dt

0

has derivative zero at s ¼ 0. The derivative of the integrand at s ¼ 0 is   1 jvj1 Kij ðtvÞvi Z j þ Zk @k Kij vj vi : 2 Integrating this expression, and then integrating by parts finds that   1  @k Kij  @i Kjk  vk vj wi ¼ 0 tv 2 for all t 2 [0, 1] and for all vectors v and w in Rn. To see what this implies, consider first taking v ¼ w. Then this says that @ t (Kij(tv)vivj) ¼ 0. Since Kij(0) ¼ 0, this implies that Kij(tv)vivj ¼ 0 for all t. This understood, take w so that wivi ¼ 0. (Thus, w is Euclidean orthogonal to v.) Then the preceding equation asserts that @t Kij ðtvÞvi wj ¼

1 k w ð@k Kij Þjtv vi vj : 2

To see what to make of this equation, differentiate with respect to s the equation   Kij tðv þ swÞ ðvi þ swi Þðvj þ swj Þ ¼ 0 and set s ¼ 0 to see that twk(@ kKij)jtvvivj ¼ 2Kij(tv)viwj. Using this fact tells us that t@t ðKij ðtvÞvi wj Þ ¼ Kij ðtvÞvi wj :

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Integration of this tells us the following: If t > t0 , then Kij(tv)viwj ¼ 0 Kij(t0 v)viwj e(tt ). Since Kij(0) ¼ 0, taking t0 ! 0 gives what we want, that Kij(tv)viwj ¼ 0 for all t, v and w. This last proposition has the following consequence: Introduce “spherical” coordinates (r, y1, . . . , yn1) for Rn. This is to say that r ¼ jxj and the coordinates {ya}1an1 are angle coordinates on Sn1. The metric in Gaussian coordinates when written in terms of these spherical coordinates for Rn has the form g ¼ dr2 þ r2 kab dya dyb where kab is a function of r and the coordinates for Sn1. Here, dr2 is a common N N shorthand for dr dr, and dya dyb is a common shorthand of dya dyb.

9.4

The proof of the geodesic theorem

The proof in what follows of the geodesic theorem (Theorem 8.1) is broken into various parts.

9.4.1

Part 1

A map, g, from an interval I  R to M is said to be locally length minimizing when the following is true: Fix any point t0 2 I and there is a neighborhood in I of t0 such that if t is in this neighborhood, then the length of g’s restriction to the interval between t and t0 is equal to dist(g(t0), g(t)). This part of the proof states and then proves a lemma to the effect that geodesics are locally length minimizing. The proof uses the Gaussian coordinates that were just introduced in the previous section. Lemma 9.3 Let p 2 M denote a given point. There exists e > 0 such that if q 2 M has distance e or less from p, then there is a unique geodesic that starts at p, ends at q and has length equal to the distance between p and q. Moreover, this is the only path in M between p and q with this length. Proof of Lemma 9.3 If q is close to p, then q will lie in a Gaussian coordinate chart centered at p. Such a chart is given by a diffeomorphism, j, from an open neighborhood, U, of p to the interior of a ball of some radius centered at 0 in Rn. Let r denote the radius of this ball. Take x ¼ (x1, . . . , xn) to be the Euclidean coordinates for Rn, thus the image of j is the ball where jxj < r. With these coordinates understood, suppose that j(q) is the point x ¼ v where 0 < jvj < r. The part of the geodesic between p and q is the path t ! x(t) ¼ tv,

9.4

The proof of the geodesic theorem

parametrized by t 2 [0, 1]. What follows explains why this path has length equal to the distance between p and q; and why it is the only such path in M. To start, suppose that g: [0, 1] ! M is any other path between p and q. Let gp denote the component of g U that starts at p as g(0). Write the j image of this N part of g in polar coordinates t ! (r(t), y1(t), . . . , yn(t)). Write the metric i dxj in polar coordinates as at the end of the preceding section. Having gij dx done so, one can write the square of the norm of g for this part of g as ¨

a b j g j2 ¼ r 2 þ r2 kab y y : It follows from this that jg j  jr j at points where g is in U, with equality if and only if g(t) ¼ f(t) v* where v* 2 Rn has norm I and where f is a smooth function on an [0, t0)  [0, 1] that contains 0. Note that this interval is a proper subset of [0, 1) if g exits the coordinate patch, in which case f(t) ! r as t ! t0. If t0 ¼ 1, then v* is the unit vector in the direction of v, and f(t) ! jvj as t ! 1. In any event, it follows from this that the length of g is greater than jvj unless the image of g is the line segment between 0 and v.

9.4.2

Part 2

Lemma 9.3 has the following corollary: A path in M whose length is the distance between its endpoints parametrizes a subset of a geodesic. The next lemma asserts this and a bit more. Lemma 9.4 Let p, q be points in M and suppose that g  M is a path from p to q whose length is the distance between p and q. Then g is an embedded closed subset of a geodesic. Proof of Lemma 9.4 It follows from Lemma 9.3 that g is part of a geodesic. As can be seen from Proposition 9.2, a geodesic is locally embedded. Thus, g is embedded unless it crosses itself. This is to say that there is a positive length subpath g0  g that starts and ends at the same point. However, if such a subpath exists, then the complement in g of the interior of g0 would be a path from p to q with length less than that of g. By assumption, no such path exists.

9.4.3

Part 3

Suppose M is a manifold with a complete Riemannian metric, g. Fix any two points, p and q, in M. This part proves that there is geodesic from p to q whose length is equal to the distance between p and q. To see why this is, suppose that L is the distance between p and q. Fix e > 0 and a map g: [0, 1] ! M with length less than L þ e and such that g(0) ¼ p and

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g(1) ¼ q. I can find N  1 (depending only on p and q) such that the following is true: There is a sequence of times 0 ¼ t0 < t1 < < tN1 < tN ¼ 1 for which the points g(tk) and g(tkþ1) for k  0 have distance 2L /N from each other and lie in a Gaussian coordinate chart centered at g(tk). I can also assume that g(tkþ2) also lies in this coordinate chart if k  N  2. It follows that the shortest path from g(tk) to g(tkþ1) is the geodesic arc between them. This understood, I can find a continuous, piece-wise smooth path from p to q with N smooth segments, with length less than L þ e, and such that each segment is a geodesic arc between its endpoints, and in particular a geodesic arc that lies entirely in a single Gaussian coordinate chart. Now, consider a sequence e1 > e2 > > em with limit zero, and for each m, a piece-wise smooth path from p to q of the sort just described. Let gm denote this path. There are N points, Pm ¼ {p1,m, . . . , pN1,m} in M where the segments that comprise gm join. This ordered set defines a point in N1 M. Thus, we have a sequence {Pm}m¼1,2, . . .  N1M. Give the latter set the metric topology from the product of the Riemannian metric topologies on the factors. The sequence {Pm}m¼1,2, . . . then lies in a bounded subset, and so in a compact subset. This is so because M is geodesically complete. This sequence has a convergent subsequence because M is geodesically complete. Let (p1, . . . , pN1) denote the limit. It follows from the way things were defined that each k  0 version of pkþ1 is contained in a Gaussian coordinate chart about pk. Here, p0 ¼ p and pN ¼ q. Likewise, pkþ2 is also contained in this chart if k  N2. The sum of the distances between pk and pkþ1 for k 2 {0, . . . , N1} is equal to L. This understood, it follows that the union of the geodesic arcs between pk and pkþ1 for k 2 {0, . . . , N1} is a length minimizing, piece-wise smooth path from p to q. To see that this curve has no kinks, use the fact that pkþ1 and pkþ2 are both in the Gaussian coordinate chart centered at pk. It follows that the shortest path between pk and pkþ2 is the arc between them, and this must then be the union of the geodesic segments between pk and pkþ1 and between pkþ1 and pkþ2. Note that the assumption of geodesic completeness is necessary. Consider, for example the case where M ¼ R2{0} with its Euclidean metric. There is no shortest path between the points (1, 0) and (1, 0) in M.

9.4.4

Part 4

Given what is said in the preceding parts of the proof, there is only one loose end left to tie so as to complete the proof of the geodesic theorem. This is the assertion that the length minimizing path can be parametrized to have constant

Additional reading

speed. Since this path is a geodesic, it is sufficient to prove that a solution to the geodesic equation has constant speed. That such is the case follows directly from Proposition 9.2. One can also prove this by writing the function t ! g(g , g ) with g a geodesic in any local coordinate chart, and differentiating the resulting expression with respect to t. Some algebraic manipulations with the geodesic equation can be used to prove that this derivative is zero.

Additional reading  A Panoramic View of Riemannian Geometry, Marcel Berger, Springer, 2003.  Riemannian Geometry, Peter Petersen, Springer, 2006.  Riemannian Manifolds: An Introduction to Curvature, John M. Lee, Springer, 1997.  Riemannian Geometry, Syllvestre Gallot, Dominique Hulin and Jacques Lafontaine, Springer, 2004.  Variational Problems in Riemannian Geometry, Seiki Nishikawa, American Mathematical Society, 2002.

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10

Principal bundles

A principal bundle is the Lie group analog of a vector bundle, and they are, in any event, intimately related to vector bundles. The definition requires the specification of a Lie group G. This chapter contains the beginnings of the principal G-bundle discussion. As was the case with vector bundles, my favorite definition is given first; the standard definition (which defines the same object) is the second definition.

10.1

The definition

Fix a smooth manifold M, and a Lie group G. A principal G-bundle is a smooth manifold, P, with the following extra data:  A smooth action of G by diffeomorphisms; thus a map m: G  P ! P with the property that m(i, p) ¼ p and m(h, m(g, p)) ¼ m(hg, p). It is customary to write this action as (g, p) ! pg1.  A surjective map p: P ! M that is G-invariant. Thus, p(pg1) ¼ p. The map p is called the projection from P to M.  Any given point in M has an open neighborhood, U, with a G-equivariant diffeomorphism j: Pju ! U  G that intertwines the map p with the evident projection from U  G to U. This is to say that if j(p) ¼ (p(p), h(p)), with h(p) 2 G, then j(pg1) ¼ (p(p), h(p)g1). Principal G-bundles P and P0 over M are said to be isomorphic when there is a G-equivariant diffeomorphism c: P ! P0 that intertwines the respective projection maps. A principal G-bundle is said to be trivial when it is isomorphic to M  G, where the latter has the obvious projection and the obvious G-action. The examples discussed here concern solely the cases where G ¼ Gl(n; R), Gl(n; C) or a matrix group such as SO(n), U(n) or SU(n).

10.2

A cocycle definition

If N  M is any subset, then PjN is used to denote p1(N)  P. If N is a submanifold, then the restriction of p and the multiplication map to PjN defines PjN as a principal G-bundle over N. Note that the third bullet of the definition asserts that any given point in M has a neighborhood, U, such that PjU is isomorphic to the product bundle. Examples of principal bundles are given momentarily.

10.2

A cocycle definition

The cocycle definition of vector bundles given in Chapter 3.2 has a principal bundle analog. This definition requires the specification of the following data: The first thing needed is a locally finite, open cover U, of M. Needed next is a collection of smooth maps from the intersections of the various sets in U to the group G; thus a set {gU,U0 : U U0 ! G}U,U 0 2 U. This data is required to obey the following conditions: ¨

U0

¨

¨

 gU,U is the constant map to the identity, i 2 G.  g1 U,U0 ¼ gU0 ,U.  if U, U0 and U00 are any three sets from U with U condition gU,U0 gU0 ,U00 gU0 0 ,U ¼ i must hold.

U00 6¼ , then the

The map gU,U0 is said to be a principal bundle transition function and the conditions listed above are called cocycle constraints. Given the data just described, the principal G-bundle is defined to be the quotient of the disjoint union UU (U  G) by the equivalence relation that puts (x, g) 2 U  G equivalent to (x0 , g0 ) 2 U0  G if and only if x ¼ x0 and g ¼ gU,U0 (x) g0 . To see that this definition is equivalent to the previous, consider first a principal bundle that is defined by this cocycle data. The group G acts so as to have h 2 G send the equivalence class of a pair (x, g) to (x, gh1). Meanwhile, the projection to M sends this same equivalence class to p. Finally, the isomorphism required by the third bullet in Chapter 10.1 is as follows: Let x 2 M. Fix a set U  U that contains x. The map to U  G of the equivalence class of U  G assigns to this class none other but the product U  G. The cocycle conditions guarantee that all of this is well defined. To go the other way, suppose that P ! M is a given principal bundle as defined in Chapter 10.1. It is a consequence of the third bullet of the definition in Chapter 10.1 that there is a locally finite cover, U, for M with the following property: If U  U, then there is an isomorphism, jU : PjU ! U  G. This understood, suppose that U and U0 are any two intersecting sets from U. 1 0 maps U Then the composition jU  jU U0 to G. Use gU,U0 to denote the latter. The collection {gU,U0 }U,U0 2U satisfies the cocycle constraints given above.

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Let PU denote the principal G-bundle as defined using this cocycle data. Define a map F: P ! PU so as to send any given point p 2 P to the equivalence class of the point jU(p) with U 2 U any set that contains p(p). This map F identifies P with PU.

10.3

Principal bundles constructed from vector bundles

What follows explains how to construct a principal bundle from a vector bundle.

10.3.1

Frame bundles

Let p: E ! M denote any given, rank n-vector bundle. Introduce PGl(E) ! M to L denote the submanifold in nE that consists of the n-tuples (e1, . . . , en) that give basis over their common base point in M. This manifold PGL(E) is a principal Gl(n; R)-bundle over M. To say more, remark that the projection to M sends a n-tuple to its base point. Meanwhile, a matrix g 2 Gl(n; R) acts to send P P 1 P 1 any given n-tuple (e1, . . . , en) to ( k g1 k1ek, k g k2ek, . . . , k g knek). To see the required isomorphisms to trivial bundles over suitable neighborhoods of points, fix attention on an open set U  M where there is a vector bundle isomorphism L L jU : EjU ! U  Rn. This isomorphism identifies ( n E)jU with U  ( n Rn) and so identifies PGl(E)jU as U  Gl(n; R) where the embedding Gl(n; R) ! n Rn sends any given n  n matrix to the n-tuple given by its n columns. By the way, the transition functions for PGl(E) are the maps to Gl(n; R) that are given by the transition functions for the vector bundle E. To see why this is, consider the open set U described in the previous paragraph, and a second open set U0  M that intersects U and comes with a trivializing isomorphism jU0 : EjU0 ! U0  Rn. Introduce gU,U0 : U U0 ! Gl(n; R) to denote the corresponding transition function for EjU U0 . Now, let (x, g ¼ (v1, . . . , vn)) 2 L U0  Gl(n; R)  U0  ( n Rn). The transition function for E identifies this point with (x, (gU,U0 (x) v1, . . . , gU,U0 ,(x) vn)) ¼ (x, gU,U0 (x)g)  U  Gl(n; R). The principal bundle PGl(E) is called the frame bundle of E. In the case when E ¼ TM, the corresponding PGL(TM) is called the frame bundle of M. ¨

¨

106

10.3.2

Orthonormal frame bundles

Suppose that E ! M is a rank n-vector bundle with a fiber metric. The metric gives the principal O(n)-bundle PO(E) ! M. This is the submanifold of PGl(E) whose n-tuples supply an orthonormal frame for E over their common base

10.3

Principal bundles constructed from vector bundles

point. To elaborate, recall from Chapter 7.1 that there is a locally finite, cover U for M with the following property: Each set U 2 U comes with an isometric isomorphism jU : EjU ! U  Rn. This isometry induces the principal bundle isomorphism jU : PO(E) ! U  O(n) by viewing PO(E) as the submanifold in L n E of n-tuples that give an orthonormal frame for E over their common base L n point while viewing O(n) as the subset of nR of n-tuples of vectors that define an orthonormal basis for Rn. Recall from Chapter 7.1 that the bundle E is said to be orientable if it can be defined using transition functions that take values in SO(n). Lemma 7.2 asserts that this is the case if and only if the real line bundle ∧n E is isomorphic to the product line bundle. In any event, if E is orientable, then there is L the analogous principal SO(n)-bundle PSO(E)  PO(E)  n E that consists of the n-tuples that define an oriented frame over their common base point. A natural case to consider arises when M has a Riemannian metric and L E ¼ TM. The bundle PO(TM) consists of the subspace in n TM of n-tuples that give an orthonormal basis for the tangent bundle to M over their common base point. The manifold M is said to be orientable when ∧n TM is the trivial line bundle. One has in this case the principal SO(n)-bundle of oriented, orthonormal frames in TM. As an example of the latter, consider the case of Sn with its round metric. The oriented orthonormal frame bundle for TSn turns out to be the group SO(n þ 1) with the projection map to Sn identifying the latter with the quotient space SO(n þ 1)/SO(n). To see how this comes about, view Sn as the unit sphere in Rnþ1. A point in the oriented, orthonormal frame bundle consists of an (n þ 1)-tuple (x, v1, . . . , vn) of orthonormal vectors in Rnþ1; where the projection to Sn gives the first element, x. The identification with SO(n þ 1) takes this (n þ 1)-tuple to the matrix with these n þ 1 vectors as its columns.

10.3.3

Complex frame bundles

Suppose that p: E ! M is a complex vector bundle with fiber Cn. Sitting inside L n E is the principal Gl(n; C)-bundle PGl(E) ! M of n-tuples that give a C-linear basis for the fiber at each point. If E has a Hermitian metric, then one can define inside PGL(E) the principal U(n)-bundle PU(E) ! M of n-tuples that define an orthonormal frame at each point. Recall in this regard Lemma 7.4 which finds such transition functions if and only if the complex rank 1 bundle ∧n E ! M is isomorphic to the product bundle M  C. What follows is an example of PU(E). View S2 as the unit radius ball about the origin in R3. Reintroduce the Pauli matrices {t1, t2, t3} from Chapter 6.4, and let E ! S2 denote the subbundle in S2  C2 of pairs (x, v) such that xjtj v ¼ iv. This bundle has its Hermitian metric from C2. Let (z, w) denote the complex

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coordinates of C2. Let p: S3 ! S2 denote the map that sends (z, w) with norm 1 to x ¼ (2re(zw ), 2im(zw ), jwj2  jzj2). The matrix xjtj is  2  jwj  jzj2 2 zw : i 2z w  jwj2  jzj2   w Note in particular that the vector v ¼ is such that xjtj v ¼ iv. Thus, z 3 PU(E) ¼ S .

10.4

Quotients of Lie groups by subgroups

The example described above with the oriented orthonormal frame bundle of Sn is but one application of a much more general construction that is summarized in the central proposition of this section. The statement of the proposition requires introducing four new notions and some notation. These are introduced in what follows directly.

10.4.1

Quotients of Lie groups

Suppose that G is a Lie group and H  G is a Lie subgroup. This is to say that H is a smooth submanifold of G that has the following three properties: First, the identity element is in H. Second, the inverse of an element in H is also in H. Finally, the product of any two elements in H is in H. For example, SO(n) is a subgroup of Gl(n; R). Likewise, SU(n) is a subgroup of U(n). If H is a subgroup of G, use G /H to denote the space of equivalence classes that are defined by the rule g g0 if and only if g0 ¼ gh for some h 2 H. The topology here is the quotient topology.

10.4.2

The Lie algebra

The Lie algebra of the group G is the tangent space to G at the identity i. This vector space is denoted by lie(G) in what follows. For example, the Lie algebra of Gl(m; R) is the vector space of n  n real matrices, and that of Gl(m; C) is the vector space of n  n complex matrices. If G ¼ SO(n), then lie(G) is the vector space A(n; R) of n  n, antisymmetric matrices. If G ¼ U(n), then lie(G) is the vector space A(n; C) of n  n, complex, anti-Hermitian matrices. If G ¼ SU(n), then lie(G) is the codimension 1 subspace in A(n; C) of trace zero matrices. The algebra designation for the vector space lie(G) signifies the existence of a bilinear map from lie(G)  lie(G) to lie(G) of a certain sort. This map is called the

10.4

Quotients of Lie groups by subgroups

Lie bracket and the Lie bracket of vectors s, t 2 lie(G) is traditionally written as [s, t]. It is such that [t, s] ¼ [s, t] and it obeys the Jacobi relation ½s; ½t; m

þ ½½s; t ; m þ ½t; ½m; s

¼ 0: In the examples in this book, G is given as a subgroup of some n  1 version of Gl(n; R) or Gl(n; C). If G is a subgroup of some n  1 version of Gl(n; R) or Gl(n; C), then lie(G) appears as a vector subspace of the corresponding vector space of real or complex n  n matrices. The Lie bracket in this case sets [s, t] ¼ st  ts. This Lie bracket can be thought of as a measure of the failure of elements in G to commute because eseteset ¼ i þ [s, t] þ e where the remainder, e, has norm that is the order of jsjjtj(jsj þ jtj). Note in this regard what is said in Chapters 8.5 and 8.6: The map s ! es sends lie(G) into G. If G is not given as a subgroup of some n  1 version of Gl(n; R) or Gl(n; C), then the Lie bracket can still be defined in the following way: Extend a given pair s, t 2 TGji as vector fields, vs and vt, that are defined in a neighborhood of the identity of G. View these as derivations of the algebra C1(G). The commutator of two derivations is also a derivation; in particular, this is the case for the commutator of vs with vt. View the latter derivation as a vector field defined near i 2 G. Its value at i in TGji is [s, t]. If G appears as a subgroup of some n  1 version of Gl(n; R), then its Lie algebra inherits the inner product that assigns the trace of sTt to a given pair s, t 2 lie(G). If G  Gl(n; C), then the inner product of a given pair s and t from lie(G) is assumed to be given by the real part of the trace of s{t. These inner products are used implicitly in what follows to define norms and orthogonal complements. If G is not given as a subgroup of one of these groups, agree beforehand to fix a convenient inner product on TGji.

10.4.3

Representations of Lie groups

Let V denote a vector space, either Rm or Cm. Introduce Gl(V) to denote either the group Gl(m; R) or Gl(m; C) as the case may be. Now suppose that G is a given Lie group. A representation of G on V is a defined to be a smooth map r: G ! Gl(V) with the property that r(i) ¼ i and r(gg0 ) ¼ r(g)r(g0 ) for any pair g, g0 2 G. Suppose that v 2 V. The stabilizer of v is the subset {g 2 G : r(g)v ¼ v}.

10.4.4

Lie algebras and representations

Suppose that r: G ! Gl(V) is a representation. Introduce r*: lie(G) ! lie(Gl(V)) to denote the differential of r at the identity. In the case when G is a matrix group, the map r* is given by the rule r* ðqÞ ¼ ðdtd rðetq ÞÞjt¼0 . This r* is a

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linear map that intertwines the Lie bracket on lie(G) with the matrix Lie bracket on lie(Gl(V)) in the sense that r*([s, t]) ¼ [r*s, r*t] for any given pair s, t 2 lie(G). That such is the case follows from the fact that r intertwines group multiplication on G with matrix multiplication. This intertwining property can be verified when G is a subgroup of some n  1 version of either Gl(n; R) or Gl(n; C) as follows: Let a, a0 2 lie(G). For any given 0 pair t, s 2 R, both eta and esa are in G. As a consequence, the assignment to (t, s) 0 0 of eta esa e-tae-sa defines a smooth map from a neighborhood of the origin in R2 to G. It follows from what was said above about the commutator that this map has the form (t, s) ! i þ ts [a, a0 ] þ r where jrj is bounded by some multiple of jtj jsj (jtj þ jsj). Given that r is a representation, it follows that 0

0

0

0

rðeta esa eta esa Þ ¼ rðeta Þrðesa Þrðeta Þrðesa Þ: Now use Taylor’s expansion with remainder to write this as i þ tsr* ð½a; a0 Þ ¼ i þ ts ½r ½a ; r* ½a0 Þ þ e * where jej is also bounded by a multiple of jtj jsj (jtj þ jsj). Given such bound for e, what is written above is possible for all t, s near 0 2 R2 only if r* intertwines the Lie bracket on lie(G) with the commutator bracket on the lie algebra of Gl(V). The stage is now set. In the proposition below, G is a lie group, V is a vector space, either Rm or Cm for some m  1 and v 2 V is a given nonzero element. Meanwhile, r: G ! Gl(V) is a representation. Proposition 10.1 The stabilizer of v 2 V is a Lie subgroup H  G whose tangent space at the identity is the subspace H ¼ {q 2 lie(G) : (r*(q))v ¼ 0}. If G is compact, then the following is also true:  The subspace Mv ¼ {r(g)v : g 2 G}  V is a smooth manifold, homeomorphic to the quotient space G / H.  The tangent space to Mv at v is canonically isomorphic to the orthogonal complement of H in lie(G) with the map that sends any given z in this orthogonal complement to r*(z)v.  The map p: G ! Mv defines a principal H-bundle. A proof of this proposition is given in Appendix 10.1; but only in the special case when G is a subgroup of some n  1 version of either Gl(n; R) or Gl(n; C).

10.5

Examples of Lie group quotients

This part of the chapter gives some examples of principal bundles that arise using the preceding proposition.

10.5

Examples of Lie group quotients

Example 10.2 This first case concerns SO(n). Use the inner product on M(n; R) given by hm, m0 i ¼ tr(mTm0 ) to view M(n; R) as a Euclidean space of dimension n2. A representation r: SO(n) ! SO(n2)  Gl(n2; R) is defined by having m 2 SO(n) act on a 2 M(n; R) via m ! mam1. Given a nonzero a 2 Mn(n; R), define the subset Ma  M(n; R) to be the set of elements of the form mam1 for m 2 SO (n). Use H  SO(n) to denote the subgroup of matrices m such that mam1 ¼ a. The proposition asserts that Ma is a smooth manifold and that the map SO(n) ! Ma defines a principal H-bundle. For a concrete example, choose a to be a diagonal matrix with distinct eigenvalues. The group H in this case is a finite group, being the group of diagonal matrices in SO(n). For a somewhat more interesting example, fix a set n1  n2  . . .  nk of positive integers that sum to n. Take a to be diagonal with m distinct eigenvalues, these of multiplicities n1, n2, . . . , nk. The subgroup H in this case is isomorphic to the subgroup S(O(n1)  . . .  O(nk))  O(n1) . . .  O(nk) consisting of the m-tuples of matrices for which the corresponding product of determinants is equal to 1. In the case when k ¼ 2, the resulting manifold Ma is diffeomorphic to the Grassmannian Gr(n; n1) of n1-dimensional subspaces in Rn, defined in Chapter 1.6. To see why this is, remark that no generality is lost by taking a to be the matrix where the top n1 elements on the diagonal are equal to 1 and the remaining are equal to 0. This understood, define a map F : Ma ! Gr(n; n1) by setting F(v) for v 2 Ma to be the span of the eigenvectors of v with eigenvalue 1. To see about an inverse, suppose that V  Rn is a given n1-dimensional subspace. Let PV denote the orthogonal projection to V. This is a self-adjoint matrix with n1 eigenvalues equal to 1 and n  n1 equal to zero. This matrix is diagonalizable, and so there exists m 2 SO(n) such that m1PVm ¼ a. Thus, PV 2 Ma. The assignment of PV to V inverts the map F. To summarize: The proposition identifies Gr(n; n1) with the manifold SO(n)/S(O(n1)  O(n  n1)). The principal bundle defined by the quotient map from SO(n) to SO(n)/S(O(n1)  O(n  n1)) can be viewed in the context of Gr(n; n1) as follows: The total space P is SO(n), and the fiber over a given subspace V  Gr(n; n1) consists of the set of matrices m 2 SO(n) that rotate Rn so as to map V to itself. In the case n1 ¼ 1, the space Gr(n; 1) is the real projective space RPn1. What is said above thus identifies RPn1 with SO(n)/S(O(1)  O(n  1)). Noting that O(1) is the two element group {1}, and noting the identification SO(n)/SO(n  1) ¼ Sn1, the proposition in this case says not much more than the fact that RPn1 ¼ Sn1/{1} where {1} acts on Rn by sending any given vector x to x.

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Example 10.3 A very much analogous example involves U(n). To say more, introduce the Hermitian inner product on M(n; C) given by hm, m0 i ¼ tr(m{m0 ) 2 to view M(n; C) as a copy of Cn . This done, the action of U(n) on M(n; C) that sends a given matrix a to mam1 defines a representation in U(n2)  Gl(n2; C). If a 2 M(n; C) is any given nonzero matrix, then the proposition finds Ma ¼ {v 2 M(n; C) : v ¼ mam1} is a smooth manifold. Moreover, the set H  U(n) of matrices m such that mam1 ¼ a is a subgroup, and the map from U(n) to Ma that sends m to mam1 defines a principal H-bundle. A concrete example takes a to be diagonal with some n1 2 {1, . . . , n1} entries equal to 1 and the remaining equal to zero. As in the SO(n) example above, the space Ma is diffeomorphic to the complex Grassmannian GrC (n; n1) as defined in Chapter 6.8. In particular, GrC(n; n1) is diffeomorphic to U(n)/(U(n1)  U(nn1)). When U(n) is viewed as a principal U(n1)  U(nn1) bundle over GrC(n; n1), its fiber over a given complex n1-dimensional subspace consists of the set of matrices in U(n) that act on Cn so as to map the subspace to itself. Note by the way, that if 1  n1   nk are integers that sum to n, then the quotient U(n)/(1mk U(m)) is the same as SU(n)/S(1mkU(m)) where S(1mkU(m)) is the subgroup of k-tuples in 1mkU(k) whose corresponding product of determinants is equal to 1. Example 10.4 This case involves SU(2) acting on C2 via matrix multiplication. Take the vector v ¼

  1 and set 0

Mv to be the set of vectors of the form mv 2 C2

with m 2 SU(2). The group H in this case is U(1) ¼ S1 and Mv ¼ S2. To be more explicit, view SU(2) as S3, the unit sphere in C2. This is done by the usual 

correspondence that pairs a given

w z

S1 as the subgroup U(1) of matrices





2 S3 with the matrix u 0

0 u 





w z

 z  w



. View

with juj ¼ 1, then the quotient

SU(2)/U(1) is S2. Thus, S2 ¼ SU(2) / U(1). This generalizes to higher dimensions as follows: View S2nþ1 as the unit sphere in Cnþ1. Then S2nþ1 ¼ U(n þ 1)/U(n) and U(n þ 1) can be viewed as the total space of a principal U(n)-bundle over over S2nþ1. Note that there is a residual U(1)-action on S2nþ1, this the action that sends a given u 2 U(1) and (z1, . . . , znþ1) 2 S2nþ1 to (uz1, . . . , uzn). This action is free, and the quotient is n-dimensional complex projective space CPn; this is the space of 1-dimensional vector subspaces in Cnþ1 from Chapter 6.8. Thus the quotient space projection, S2nþ1 ! CPn, defines a principal U(1)-bundle. Another view of this bundle is given in Example 10.9 in the next section.

10.6

10.6

Cocycle construction examples

Cocycle construction examples

What follows describes some principle bundles that are constructed using the cocycle definition in Chapter 10.2. Example 10.5 Principle U(1)-bundles over S2 can be constructed using the cocycle definition as follows: View S2 as the set of unit length vectors in R3, and this done, decompose it as S2 as Uþ ¨ U where Uþ consists of vectors x ¼ (x1, x2, x3) with jxj ¼ 1 and x3 > 1, and where U consists of vectors with norm 1 and with x3 < 1. These two sets intersect in the cylindrical region S2 that consists of the points in R3 with norm 1 and with x3 2 (1, 1). This understood, the complex number z ¼ x1 þ ix2 is nowhere zero on this region, and so the assignment x ! jzj1 z defines a smooth map from Uþ U into the unit circle in C, thus the group U(1). Let g: Uþ U ! U(1) denote this map. Now, let m denote any given integer. A principal U(1)-bundle is defined by taking for cocycle data the open cover U ¼ {Uþ, U} and principal bundle transition function gm: Uþ U ! U(1). There is an analogous construction for any given surface. Let  denote the surface in question. Fix a finite set    of distinct points. Let p 2 . Fix an open set Up   containing p and with a diffeomorphism jp : Up ! R2 that sends p to the origin. Do this for each point in , but make sure to choose the sets in question so that their closures are pair-wise disjoint. Assign an integer, m(p) to each p 2 . The cover U ¼ {U0 ¼  –} ¨ {Up}p2 is used for the cocycle data set. A bundle transition function g0p : U0 Up ! U(1) is defined as follows: If x 2 U0 Up, write jp(x) as a complex number, z(x). This done, set g0p(x) ¼ z(x)/jz(x)jm(p). As no three distinct sets from the cover intersect, this data defines a principal U(1)-bundle over . ¨

¨

¨

¨

¨

Example 10.6 Principal SU(2)-bundles over S4 can be constructed as follows: Identify the equatorial S3 as SU(2). Now define a principal bundle SU(2) over S4 by declaring that it be isomorphic to the trivial bundle over the complement of the south pole, and also trivial over the complement of the north pole. Use U to denote the former set and V to denote the latter. The intersection of these two sets can be described as follows: View S4 as the set x ¼ (x1, . . . , x4, x5) 2 R5 with jxj ¼ 1. Then U is the set x5 > 1 and V is the set x5 < 1. Thus, their overlap is the set where x5 6¼ 1. Define a map from this set to S3 by sending y to the point g(x) ¼ (x1, x2, x3, x4)/(1  x52 )1/2. Identify S3 with SU(2) as done previously. Fix an integer p, and define a principal SU(2)-bundle P(p) ! S4 by declaring its transition function to be g(x)p. As it turns out the bundle P(1) is diffeomorphic to S7. So is P(1), but these bundles are not isomorphic as principal SU(2)-bundles.

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To see the relation to S7, it is useful to introduce H to denote the set of 2  2 complex matrices that can be written as    a b :  b a If I reintroduce the Pauli matrices {t1, t2, t3} from Chapter 6.4, then an element in H can be written as x4 þ x1t1 þ x2t2 þ x3t3. Here, a ¼ x4 þ ix3 and b ¼ x2 þ ix1. Thus, H can be viewed as R4, or as C2. Given the multiplication rule described in Chapter 6.4, this last way of writing H identifies it with the vector space (and algebra via matrix multiplication) of quaternions. In any event, the 7-sphere is the set of pairs (a, b) 2 H  H with jaj2 þ jbj2 ¼ 1. Here, jaj2 ¼ tr(a{a). The group SU(2) sits in H as the set of unit vectors. It acts on S7 so that m 2 SU(2) sends (a, b) to (am1, bm1). It can also act to send (a, b) to (ma, mb). The quotient of either action is S4, and one quotient defines P(1), the other defines P(1). To see that the quotient is S4, define a map from S7 to R5 by sending a given vector (a, b) to the vector (2ab{, jaj2  jbj2) 2 H  R ¼ R4  R ¼ R5. This vector has unit length, so defines a point in S4. This map is constant along orbits of SU(2) by the action that sends m 2 SU(2) and (a, b) 2 S7 to (am1, bm1). As exhibited momentarily, it is the projection to S4 that defines the principal bundle P(1). The following description of the cocycle data for P(1) verifies that the map just given from S7 to S4 is indeed the principal bundle projection. To start, identify Uþ with H via the map that sends any given point z 2 H to (2z, 1  jzj2)/(1 þ jzj2). Define the inverse map from H  SU(2) to S7 so as to send (z, m) to (a ¼ zm, b ¼ m)/(1 þ jzj2)1/2. Over the set U, define a map from H to U by sending r 2 H to (2r{, jrj2  1)/(1 þ jrj2)1/2. Now define a map from H  SU(2) to S7 by sending (r, n) to (a ¼ n, b ¼ rn)/(1 þ jrj2)1/2. The equator is the set jzj ¼ jrj ¼ 1 and r ¼ z{. Thus, the points m and n are sent to the same point if n ¼ zm. This is the transition function for P(1). There is a very much analogous story for P(1) that identifies the bundle projection with the map from S7 that sends (a, b) to (2a{b, jaj2  jbj2). Here is a parenthetical remark: One can define a fiber bundle over S4 with fiber SU(2) (but not a principal bundle) as follows: Choose integers (p, q) and declare the space Xp,q to be diffeomorphic over Uþ to Uþ  SU(2) and also diffeomorphic over U to U  SU(2). Over the intersection of these sets, define the transition function so as to send (x, m) 2 Uþ  SU(2) to U  SU(2) using the rule (x, m) ! (x, g(x)pmg(x)q). This is a principal bundle if and only if one of p or q is zero. Now restrict to the case where p þ q ¼ 1, and set k ¼ p  q. John Milnor proved that Xp,q in this case is homeomorphic to S7 but not diffeomorphic to S7 if k2  1 6¼ 0 mod(7). In fact, each residue class mod(7) gives a different smooth structure on S7. As it turns out, there are 28 different

10.6

Cocycle construction examples

smooth structures on S7. All can be realized as follows: Intersect the unit sphere about the origin in C5 with the submanifold in C5{0} where the complex coordinates (z1, . . . , z5) obey the equation ¼0 z21 þ z22 þ z23 þ z34 þ z6k1 5 for k 2 {1, 2, . . . , 28}. (These are known as Brieskorn spheres.) Note that this submanifold is a complex manifold, and that it intersects S9 transversely. To read more about exotic spheres, see John Milnor’s original article, “On manifolds diffeomorphic to the 7-sphere”, Annals of Mathematics 64 (1956) 399–405, and the paper by John Milnor and Michel Kervaire, “Groups of homotopy spheres I”, Annals of Mathematics 77 (1963) 504–537. See also the entry by J.P. Levine, “Lectures on groups of homotopy spheres” on pages 62–95 of the book Algebraic and Geometric Topology, Lecture Notes in Mathematics 1126, Springer, 1985. Example 10.7 Let M denote any given 4-dimensional manifold. Choose a finite set, , of distinct points in M and associate to each an integer. For each p 2 , fix a coordinate chart U, centered at p, with coordinate map j: U ! R4. Take U so that charts given by distinct points from  are disjoint. Define a principal SU(2)-bundle P ! M as follows: Set PjM to be isomorphic to (M)  SU(2). For each p 2 , set PjU ¼ U  SU(2) also. To define the transition functions, first agree to identify the unit sphere in R4 with S3. This done, let q denote the integer associated to the point p, and then the transition function, gU,M to be the map that sends x 2 U (Mp) ¼ Up to (j(x)/jj(x)j)q 2 SU(2). ¨

Example 10.8 Something along the lines of what is done in Example 10.7 can be done any time you have an interesting map from Sn1 to SO(k), U(k), or some other group G. Let f denote your map. Pick a set  of distinct points in Mn, and associate to each an integer. Choose for each point a coordinate chart as in the previous example. Define the bundle to be trivial over M and over each coordinate chart. If p 2 , the transition function for U (Mp) ¼ Up sends x to f(x)q. For example, recall from Chapter 6.6 the definition of the matrices {ga}a=1, . . . ,5 in M(4; C). Define a map from R5 to M(4; C) by sending P a given point x ¼ (x1, . . . , x5) to f(x) ¼ j xjgj. Note that f {f ¼ jxj2, so f maps S4 to U(4). ¨

Example 10.9 This example returns to the principal U(1)-bundle that is defined by the projection from S2nþ1 to CPn as described at the end of the previous part of this chapter. Recall from Chapter 6.8 that CPn has an open cover by n þ 1 sets with each set in the cover, U ¼ {O 1, . . . , O nþ1} diffeomorphic to Cn. By way of reminder, any given point (z1, . . . , zn) 2 O k ¼ Cn parametrizes the complex line in Cnþ1 that is spanned (over C) by the vector with k’th entry is 1, whose i’th entry for i < k is zi, and whose i’th entry for i > k is zi1.

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This understood, the intersection between O k and O k0 consists of the complex lines in Cnþ1 spanned by vectors of the form (p1, . . . , pnþ1) with both pk and pk0 nonzero. A map from O k O k0 to U(1) assigns to any such line the complex number gk,k0 ¼ jpkj1jpk0 jpk/pk0 . The collection {gk,k0 }1knþ1 defines principal U(1)-bundle transition functions for the open cover U. To see that such is the case, it is enough to verify that the cocycle condition is obeyed on any given triple intersection. To do this, suppose that k, k0 , and k00 are distinct. Then gk,k0 gk0 ,k0 0 gk00 ,k is observedly 1 as required. By the way, the identification between the trivial bundle O k  U(1) with a subset of S2nþ1 is as follows: Use z ¼ (z1, . . . ., zn) as before for the coordinates on O k. Let u 2 U(1). The point (z, u) 2 O k  U(1) gives the point whose k’th coordinate is (1 þ jzj2)1/2u, whose i’th coordinate for i < k is (1 þ jzj2)1/2ziu, and whose i’th coordinate for i > k is given by (1 þ jzj2)1/2zi1u. ¨

116

Example 10.10 This example constructs what are known as quaternionic projective spaces. To start, fix a positive integer, n. Consider R4(nþ1) as (nþ1) H. Let S4nþ3 denote the unit sphere in this Euclidean space; and define an action of SU(2) on S4nþ2 by sending any given m 2 SU(2) and (a1, . . . , anþ1) 2 S4nþ3 to (a1m1, . . . , anþ1m1). The space of orbits, S4nþ3/SU(2) is a smooth manifold which is usually denoted by HPn, the quaternionic projective space. It has dimension 4n. The quotient map defines a principal SU(2)-bundle. The proof that such is the case copies in an almost verbatim fashion what is said in the preceding example about the principle U(1)-bundle S2nþ1 ! CPn. The details are left to you the reader to work out.

10.7

Pull-backs of principal bundles

Let N and M denote a pair of smooth manifolds, and let f: M ! N denote a smooth map. As explained in Chapter 5.1, the map f can be used to pull-back a vector bundle over N to obtain a vector bundle over M. Principal bundles can also be pulled back using f. To elaborate, suppose that G is a compact Lie group and p: P ! N is a principal G-bundle. The pull-back bundle f*P is a principal bundle over M that is defined as follows: It sits in M  P as the subset of pairs (x, p) with f(x) ¼ p(p). Here is a neat way to see f*P as a manifold: Let N  N  N denote the diagonal, thus the set of pairs (y, y) with y 2 N. This is a submanifold. Map M  P to N  N using the map that sends a pair (x, p) to (f(x), p(p)). It is a straightforward exercise to verify that this map is transversal to the diagonal in the sense used in Chapter 5.7. This understood, the proposition in Chapter 5.7 guarantees that the inverse image of N in M  P is a submanifold. This inverse image is f*P.

10.7

Pull-backs of principal bundles

To see that f*P meets the principal bundle requirements listed in Chapter 10.1, note to start that G acts so that any given g 2 G sends a point (x, p) 2 f*P to (x, pg1). Meanwhile, the map p to M sends (x, p) to x. The last of the three requirements listed in Chapter 10.1 is the following: Given x 2 M, needed is an open set, U  M, that contains x and comes with a G-equivariant diffeomorphism from PjU to U  G that intertwines p with the projection from U  G to U. To obtain this data, go to N and select an open set V  N that contains f(x) and comes with a G-equivariant diffeomorphism, jV, from PjV to V  G that intertwines the map from PjV with the projection to V. Let cV: PjV ! G denote the composition of first jV and then projection to G. Now take U ¼ f1(V) and set jU : f*PjU ! U  G to be the map that sends a pair (x, p) to (x, cV(p)). It is perhaps redundant to say that the bundle f*P comes with the tautological, G-equivariant map to P, this the map that sends a given pair (x, p) to p. Example 10.11 Chapter 10.1 declared a pair of principal G-bundles p: P ! M and p0 : P0 ! M to be isomorphic if there exists a G-equivariant diffeomorphism f : P ! P0 such that p ¼ p0  f . Keeping this in mind, suppose that p : P ! M defines a principal U(1)-bundle. As it turns out, there exists an integer n  1, a map f: M ! CPn, and an isomorphism between P and the pull-back f*S2nþ1 with S2nþ1 viewed here as a principal U(1)-bundle over CPn; this as described in Example 10.9. What follows gives a construction of this data when M is compact. To start, fix a finite cover, U, of M such that each set U from U comes with a bundle isomorphism jU : PjU ! U  U(1). Let cU : P ! U(1) denote the composition of first jU and then the projection to the U(1) factor. By refining this cover, I can assume that each open set is also a coordinate chart, and so there is a subordinate partition of unity. Recall from Appendix 1.2 that this is a collection, {wU}U2U, of nonnegative functions such that any given U 2 U P version of wU has support only in U, and such that U2U wU ¼ 1 everywhere. Set hU( ) to denote the function on P that is obtained by composing p with P wU( U2U wU2)1/2. Write the number of sets in the cover U as n þ 1, and label the sets in U from 1 to n þ 1. Define now a map, ^f, from P to S2nþ1  Cnþ1 by taking the k’th coordinate to be zero on the complement of Uk and equal to hUk cUk on Uk. This map is U(1)-equivariant, and so the quotient of the image by U(1) defines a map, f, from M to CPn. The bundle isomorphism between P and f*S2nþ1 sends p 2 P to (p(p), ^f(p)) 2 M  S2nþ1. Example 10.12 Example 10.10 describes the quaternionic projective spaces {HPn}n1; and it explains how to view the map from S4nþ3 to HPn as a principal SU(2)-bundle. Let p: P ! M denote a given principal SU(2)-bundle. A construction that is very much like that in the preceding example will give

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an integer, n  1, a map f: M ! HPn, and a principal bundle isomorphism between P and the pull-back bundle f*S4nþ3. Example 10.13 Let N denote a smooth manifold and suppose that E ! N is a vector bundle with fiber dimension n, either real or complex. As explained in Chapter 10.3, one can construct from E the principal bundle of frames for E. This is a principal bundle with group G either Gl(n; R) or Gl(n; C) as the case may be. If E also has a fiber metric, one can use the metric to construct the principal G ¼ O(n)-, SO(n)- , U(n)- or SU(n)-bundle depending on the circumstances. I use PE in what follows to denote any one of these and use G to denote the relevant Lie group. Now let M denote a second smooth manifold and suppose that f: M ! N is a smooth map. As explained in Chapter 5.1, the map f can be used to pull E back so as to give a vector bundle, f*E, over M. Metrics on bundles pull-back as well, the norm of a vector (x, v) 2 f*E  M  E being jvj. If E has a metric, then give f*E this pull-back fiber metric. In any event, one has the principal G-bundle Pf*E over M. Meanwhile, principal G-bundles pull-back as explained above, so one also has the principal G-bundle f*PE. I hope it is no surprise that Pf*E is the same as f*PE. Those who doubt can unravel the definitions and so verify the assertion.

10.8

Reducible principal bundles

Suppose that G is a Lie group, H  G is a subgroup, and P ! M is a principal H-bundle. A principal G-bundle over M can be constructed from P as follows: PG ¼ ðP  GÞ=H ¼ PH G; where the equivalence relation identifies pairs (p, g) with (ph1, hg). Multiplication by an element g 2 G sends the equivalence class (p, g) to that of (p, gg1). To see that this is a principal bundle (and a manifold), go to any given set U  M where PjU admits an isomorphism, j, with U  H. This isomorphism identifes PG with U  (H  G) /H which is U  G. If gU,U0 : U U0 ! H is the transition function for P over an intersection of charts, then this is also the transition function for PG with the understanding that H is to be viewed as a subgroup in G. A principal G-bundle P0 ! M that is isomorphic to P H G with P ! M some principal H-bundle and with H  G a proper subgroup is said to be reducible. For example, if E ! M is a complex vector bundle with fiber Cn with Hermitian metric, then one can construct the U(n) principal bundle PU(E) ! M. This bundle is reducible to a principal SU(n)-bundle if the vector bundle ∧n E ¼ det(E) is trivial. ¨

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10.9

Associated vector bundles

Every bundle has a fiber metric, and this implies that every principal Gl(n; R)-bundle is reducible to a principal O(n)-bundle. (If the transition functions can be chosen so as to have positive determinant, then the bundle is reducible to an SO(n)-bundle.) By the same token, any given principal Gl(n; C)bundle is reducible to a principal U(n)-bundle. This is because the metric can be used to find local trivializations for the bundle whose transition functions map to O(n) or U(n) as the case may be. Here is another example: If a 2n-dimensional manifold M admits an almost complex structure, j, then its principal Gl(2n; R) frame bundle is reducible to a Gl(n; C)-bundle. Here, Gl(n; C)  Gl(2n; R) is identified with the subgroup, Gj, of matrices m such that mj0 ¼ j0m where j0 is the standard almost complex structure on R2n as defined in Chapter 2.4.

10.9

Associated vector bundles

Chapter 10.3 explains how to construct a principal bundle from a given vector bundle. As explained here, this construction has an inverse of sorts. To set the stage, suppose that G is a Lie group and p: P ! M is a principal G-bundle. The construction of a vector bundle from P requires as additional input a representation of the group G into either Gl(n; R) or Gl(n; C). Use V here to denote either Rn or Cn as the case may be; and use Gl(V) to denote the corresponding general linear group, either Gl(n; R) or Gl(n; C). Let r denote the representation in question. Recall from Chapter 10.4 that r is a smooth map from G to Gl(V) two special properties: First, r(i) is the identity in the matrix group Gl(V). Second, r(gg0 ) ¼ r(g)r(g0 ) for any given pair of elements g, g0 in G. The corresponding vector bundle has fiber V and is denoted by P r V. It is the quotient of P  V by the equivalence relation that equates a given pair (p, v) to all pairs of the form (pg1, r(g)v) with g 2 G. Said differently, G acts on P  V so that any given g 2 G sends any given pair (p, v) to (pg1, r(g)v). The space P r V is the space of G-orbits via this action. To verify that this is indeed a vector bundle, note first that the projection to M is that sending the equivalence class of a given (p, v) to p(p). The action of R or C has a given real or complex number z sending (p, v) to (p, zv). The zero section, oˆ, is the equivalence class of (p, 0) 2 P  V. To see the local structure, let x 2 M denote any given point and let U  M denote an open set with a principal bundle isomorphism, j, from PjU to U  G. Use c: P ! G to denote the composition of first j and then projection from U  G to G. Now define jV: (P r V)jU ! U  V to be the map that sends the equivalence class of (p, v) to (p(p), r(c(p)) v). Note that this map is 1–1 and invertible; its inverse sends (x, v) 2 U  V to the equivalence class of (j1(x, i), v).

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This same map jV defines the smooth structure for (P r V)jU. I claim that this smooth structure is compatible with that defined for (P r V)jU0 when U and U0 are intersecting open sets. To verify this claim, let j: PjU ! U  G and j0 : PjU0 ! U0 G denote the associated principal bundle isomorphisms. The composition j0  j1 maps (U0 U)  G as a map, (x, g) ! (x, gU0 ,U(x) g) 0 0 where gU ,U maps U U to G. The corresponding vector bundle transition function j0 V jV sends (x, v) to (x, r(gU0 ,U(x)) v), and this is, of course, a smooth, invertible map that is linear over R or C as the case may be. What follows are examples of associated vector bundles. ¨

¨

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Example 10.14 If E ! M is a given vector bundle, with fiber V ¼ Rn or Cn, then I defined in Chapter 10.3 the bundle PGl(E) which is a principal G ¼ Gl(n; R) or G ¼ Gl(n; C)-bundle as the case may be. Let r now denote the defining representation of G on V. Then PGl(E) r V is canonically isomorphic to E. Of course, they have the same transition functions, so they are isomorphic. L Another way to see this is to view PGL(E)  n E as the set of elements (e1, . . . , en) that span the fiber of E at each point. Define f: (PGl(E)  V) ! E P by sending e ¼ (e1, . . . , en) and v to f(e, v) ¼ j vjej. This map is invariant with respect to the G-action on PGl(E)  V whose quotient gives PGl(E) r V, and so the map descends to the desired bundle isomorphism between PGl r V and E. If V ¼ Rn and E has a fiber metric, or if V ¼ Cn and E has a Hermitian metric, I defined in Chapter 10.3 the respective principal SO(n)- or U(n)-bundles PO(E) and PU(E). Let r denote the standard inclusion homomorphism from SO(n) into Gl(n; R) when V ¼ Rn, and let it denote the corresponding homomorphism from U(n) into Gl(n; C) when V ¼ Cn. Reasoning as in the previous paragraph leads to the conclusion that PO(E) r Rn ¼ E in the real case, and PU(E) r Cn ¼ E in the complex case. Example 10.15 This example describes a sort of converse to what is said in the previous example. To set the stage, let V ¼ Rn or Cn. In the real case, use G to denote either Gl(n; R) or O(n); and in the complex case, use G to denote either Gl(n; C) or or U(n). In any case, use r to denote the standard representation of the relevant group acting on V. Suppose that P ! M is a principal G-bundle, and let E ¼ P r V. In the real case, P is then isomorphic to PGl(E) or PSO(E) as the case may be. In the complex case, P is isomorphic to PGL(E) or PU(E). Now recall from Chapter 5.2 in the real case that there exists m > n and a smooth map from M to the Grassmannian Gr(m; n) with the property that its pull-back of the tautological Rn-bundle is isomorphic to E. Chapter 6.11 tells a similar story in the complex case: There exists m > n and a map from M to GrC (m; n) whose pull-back of the tautological Cn-bundle is isomorphic to E. Keeping these facts in mind, let Gr denote Gr(m; n) in the real case and GrC (m; n) in the complex case, and let f : M ! Gr denote the relevant map.

Appendix 10.1

Proof of Proposition 10.1

Introduce ET ! Gr to denote the aforementioned tautological G-bundle, and use PT ! Gr to denote the bundle of frames in ET. Take these to be orthonormal or unitary in the case that G is SO(n) or U(n) respectively. The map f also pulls back PT, and f *PT ¼ PGl(f *ET) or PO(f *ET) or PU(f *ET) as the case may be. It follows as a consequence that the original bundle P is isomorphic to the appropriate version of f *PT. Here is a formal statement of this last observation: Proposition 10.16 Let G denote either Gl(n; R) or SO(n) in the real case, and either Gl(n; C) or U(n) in the complex case. Let M denote a smooth manifold and let P ! M denote a principal G-bundle. There exists m  n and a smooth map from M to the Grassmannian Gr whose pull-back of the principal G-bundle PT ! Gr is isomorphic to P. Example 10.17 The neat thing about this associated vector bundle construction is that all bundles that are constructed from E via various algebraic operations on the fiber vector space V ¼ Cn or Rn, such as direct sum, tensor product, dualizing, Hom, etc., can be viewed as arising from PGl(E) through N p p an associated bundle construction. For example, n E, ∧ E, Sym (E), E* ¼ Hom(E; R or C), etc. correspond to representations of Gl(n; R) or Gl(n; C) into various vector spaces. Thus, all of the latter bundles can be studied at once by focusing on the one principal bundle PGl(E). One example is as follows: Let g be a Riemannian metric on a manifold M, and so that there exists the principal O(n)-bundle PO(TM) ! M of oriented, orthonormal frames in TM. Let r denote the representation of O(n) on the vector space A(n; R) of n  n antisymmetric matrices that has g 2 O(n) act so as to send a 2 A to r(g)a ¼ gag1. This representation is called the adjoint representation, and the associated vector bundle is isomorphic to ∧2 T*M. A second example along these same lines concerns the bundle ∧n T*M. The latter is associated to PO(TM) via the representation of O(n) in the two group O(1) ¼ {1} that sends a matrix g to its determinant. Vector bundles that are associated to the orthonormal frame bundle of a Riemannian manifold are called tensor bundles, and a section of such a bundle is said to be a tensor or sometimes a tensor field.

Appendix 10.1

Proof of Proposition 10.1

Assume below that G is a subgroup of some n  1 version of either Gl(n; R) or Gl(n; C). To see that H is a subgroup, note that if h and h0 are in H, then r(hh0 )v is the same as r(h)r(h0 )v ¼ r(h)v ¼ v. To see that h1 2 H, note that r(i)v ¼ v, and because i ¼ h1h, so r(h1h)v ¼ v. As r(h1h) ¼ r(h1)r(h), this

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means that r(h1)r(h)v ¼ v. If r(h)v ¼ v, then this last identity requires that r(h1)v ¼ v also. To see that H is a submanifold and a Lie group, it is sufficient to give a local coordinate chart near any given element. To start, remark that a chart for a neighborhood of the identity i supplies one for a neighborhood of any other element in H. Here is why: If U  H is an open set containing i with a diffeomorphism j: U ! Rm for some m, then the set Uh ¼ {h0 : h0 h1 2 U} is a chart near h. This understood, the diffeomorphism jh that sends h0 to j(h0 h1) gives the desired diffeomorphism for Uh. Granted all of this, what follows constructs coordinates for some neighborhood of i. To do this, introduce H  lie(G) to denote the kernel of the differential of r at the identity. This is a Euclidean space whose dimension depends, in general, on v. Let m denote this dimension. As is argued next, the exponential map h ! eh restricts H so as to map the latter into H. For a proof, fix a large integer N >> 1 and write eh ¼ eh/N . . . eh/N as the product of N factors. This allows r(eh)v to be written as r(eh/N) . . . r(eh/N)v. Meanwhile, Taylor’s theorem with remainder finds that rðeh=N Þv ¼ rðiÞv þ

1 r ðhÞv þ e1 ¼ v þ e N *

where jej  c0 N2. This understood, it follows by iterating this last result N times that rðeh Þv ¼ v þ eN where jeNj  c0 N1. As N can be as large as desired, this means that r(eh)v ¼ v. It follows from what was just said that the map h ! eh embeds a ball about 0 in H into H. What follows argues that this embedding restricts to some ball about 0 in H so as to map onto a neighborhood of i. Note in the meantime that this result implies that the map h ! eh gives local coordinates for a neighborhood of the identity in H. It also implies that H is a totally geodesic submanifold of G when given the metric that is induced from the left invariant metric on G that comes from the latter’s identification as a submanifold of Gl(n; R) or Gl(n; C) as the case may be. To see that this map h ! eh is onto a neighborhood of the identity, suppose h 2 H is near the identity. As noted in Chapter 8.7, this element h can be written in any event as em for some m 2 lie(G). Write m ¼ h þ z with h 2 H and with z orthogonal to H and jmj small. Keep in mind that with jmj being small, then both jhj and jzj are small as both are less than jmj. This understood, note that ehþz ¼ ehez þ e where jej  c0 jhj jzj. It follows as a consequence, that r(ehþz)v ¼ r(ez)v þ v0 with jv0 j  c0 jhj jzj. Meanwhile,

Appendix 10.1

Proof of Proposition 10.1

rðez Þ ¼ i þ r* ji z þ Oðjzj2 Þ and so r(ez)v ¼ v þ (r*jiz) v þ v00 where jv00 j  c0 jmj jzj. Given that z is orthogonal to H, it follows that jr(ez)v  vj  c01jzj jvj when jmj is small. Given that h is in H, so it follows that z ¼ 0 if h is in some neighborhood of i. A suitable cover of Mv by charts will give it a manifold structure. Keep in mind when defining the charts that any given element in Mv is mapped to v by the action of G on V. As a consequence, it is sufficient to construct a local chart near v. Indeed, if U  Mv is a neighborhood of v with a homeomorphism j: U ! Rd for some d, and if v0 ¼ r(g)v for some g 2 G, then Uv0 ¼ {b 2 Ma : r(g1)b 2 U} is a neighborhood of v0 . Moreover, the map jv0 that sends b 2 Uv0 to j(r(g1)b) is then a local homeomorphism from Uv0 to Rd. Let H⊥  lie(G) denote the orthogonal complement of H. The chart for a neighborhood of v with its homeomorphism is obtained by defining the inverse, this a homeomorphism, c, from a ball in H⊥ to a neighborhood of v that sends the origin to v. This homeomorphism sends z 2 H⊥ to c(z) ¼ r(ez)v. This map 0 is 11 on a small ball about the origin. This because ez  ez ¼ z  z0 þ r with jrj  c0(jvj þ jv0 j) jv  v0 j . Thus, 0

rðez Þ  rðez Þ ¼ r* ji z  r* ji z0 þ r0 0

where jr0 j  c0 ðjvj þ jv0 jÞjv  v0 j :

As a consequence, r(ez)v ¼ r(ez )v if and only if z  z0 2 H when both jzj and jz0 j are small. This is possible only if z  z0 ¼ 0. To see why this map is onto a neighborhood of v in Mv, suppose for the sake of argument that there is a sequence {vk}k1  Mv that converges to v, but is such that no element from this sequence can be written as r(ez)v with z 2 H⊥ having small norm. By definition, each vk can be written as r(gk)v with gk 2 G. Moreover, there exists d > 0 such that each gk has distance at least d from any point in H. Here, the metric is that used in Chapter 8.7. To see why this is, note that if such were not the case, then gk could be modified by multiplying on the right by the closest element in H so that the result is close to the origin in G. The result could then be written as exp(z)exp(h) with h 2 H and z 2 H⊥; and this would exhibit vk as r(ez)v. Meanwhile, the fact that G is compact implies that the sequence {gk}k1 has a convergent subsequence. Let g denote the limit. Given that the elements in the sequence {gk} are uniformly far from H, this matrix g is not in H. Even so r(g)v ¼ v. As these last two conclusions contradict each other, there is no sequence {vk}k1 as described above. Note that this is the only place where the compactness of G is used. To finish the proof of Proposition 10.1, consider the claim in the last bullet of the proposition. This claim follows with a proof that the map from G to Mv has the property that is described by the third bullet in Chapter 10.1’s definition of a principal bundle. This understood, note that it is sufficient to verify the

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following: There is an open set U  Mv containing v with a diffeomorphism j: GjU ! U  H that is suitably equivariant. To prove that such is the case, fix a small radius ball B  H⊥ on which the map z ! r(ez)v is a diffeomorphism onto its image. Let U denote the image of this diffeomorphism, and set c: U ! B to be the inverse of this diffeomorphism. Thus, r(ec(y))v ¼ y for all y 2 U. Define j on GjU to be the map that sends g 2 GjU to j(g) ¼ r(g)v, ec(r(g))g). This gives the required H-equivariant diffeomorphism.

Additional reading  Topology of Fibre Bundles, Norman Steenrod, Princeton University Press, 1999.  Fibre Bundles, Dale Husemo¨ller, Springer, 1993.  Basic Bundle Theory and K-Cohomology Invariants, Dale Husemoller, Michael Joachim, Branislaw Jurco and Marten Schottenloher, Springer, 2009.  Metric Structures in Differential Geometry, Gerard Walschap, Springer, 2004.  Differential Geometry, Lie Groups and Symmetric Spaces, Sigurdur Helgason, American Mathematical Society, 2001.

11

Covariant derivatives and connections

Let p: E ! M denote a vector bundle, with fiber V ¼ Rn or Cn. Use C1(M; E) to denote the vector space of smooth sections of E. Recall here that a section, s, is a map from M to E such that p  s is the identity. This is an infinite dimensional vector space. The question arises as to how to take the derivative of a section of E in a given direction. Of course, one can take the differential of a map, this giving a linear map s*: TM ! s*TE. This notion of derivative turns out to be less than satisfactory. Needed is a derivative that defines a bundle homomorphism from TM to E, not to s*TE. At issue here is how to define such a derivative. As explained in what follows, there are various ways to proceed, though all give the same thing at the end of the day. One such definition involves principal Lie group bundles and the notion of a connection on such a bundle. The related notions of covariant derivative and connection are the focus of this chapter.

11.1

Covariant derivatives

The space C1(M; E) is a module for the action of the algebra, C1(M), of smooth functions with values in R if V ¼ Rn or values in C if V ¼ Cn. The action is such that a given function f acts on s 2 C1(M; E) to give fs. A covariant N derivative for C1(M; E) is a map : C1(M; E) ! C1(M; E T*M) with the following two properties: First, it respects the vector space structure. This is to say that (cs) ¼ cs and (s þ s0 ) ¼ s þ s0 and when c is in R or C (as the case may be), while s and s0 are sections of E. Second, it obeys the analog of Leibnitz’s rule: ðfsÞ ¼ fs þ s for all f 2 C1(M).

O

df

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To see that there exist covariant derivatives, it is enough to exhibit at least one. To obtain one, take a locally finite open cover, L, of M such that any given open set U 2 L comes with a vector bundle isomorphism jU : EjU ! U  V. Let {wU}U2L denote a subordinate partition of unity. Thus, wU : M ! [0, 1] has P support only in U, and U2L wU ¼ 1 at each point. Save this partition for a moment. For each U 2 L, define the covariant derivative, d, on C1(U; UV) as follows: Write a given section, v, of the trivial bundle U  V as a map x ! (x, v(x)). Now define the covariant derivative dv as the map x ! (x, dvjx). Now, suppose that s is any given section of E. Define s ¼ ∑U2L wU j* U ðdðjU  sjU ÞÞ: Note that this obeys the Leibnitz rule by virtue of the fact that

11.2

P U2L

wU ¼ 1.

The space of covariant derivatives

There are lots of covariant derivatives. As is explained next, the space of covariN ant derivatives is an affine space modeled on C1(M; Hom(E; E T*M)). To see N this, first note that if a 2 C1(M; Hom(E; E T*M)) and if  is any given covariant derivative, then  þ a is also a covariant derivative. Meanwhile, if  and 0 are both covariant derivatives, then their difference,   0 , is a section of N T*M). This is because their difference is linear over the action of Hom(E; E C1(M). The following lemma puts this last remark in a more general context. N Lemma 11.1 applies to the situation at hand using for E0 the bundle E T*M and taking L ¼ 0  . Lemma 11.1 Suppose that E and E0 are vector bundles (either real or complex) and that L is an R- or C-linear map (as the case may be) that takes a section of E to one of E0 . Suppose in addition that L(f s) ¼ f L(s) for all functions f. Then there exists a unique section, L, of Hom(E; E0 ) such that L( ) ¼ L( ). Proof of Lemma 11.1 To find L, fix an open set U  M where both E and E0 has a basis of sections. Denote the basis for E and the basis for E0 as {ea}1ad and {eb0 }1bd0 where d and d0 denote here the respective fiber dimensions of E and E0 . Since {eb0 }1bd0 is a basis of sections of E0 , any given a 2 {1, . . . , d} version of Lea can be written as a linear combination of this basis. This is to say that there are functions {Lab}1bd0 on U such that Lea ¼ ∑1bd0 Lab e0b:

11.3

Another construction of covariant derivatives

This understood, the homomorphism L is defined over U as follows: P Let s denote a section of E over U, and write s ¼ 1ad sa ea in terms of the basis {ea}1ad. Then Ls is defined to be the section of E0 given by P Ls ¼ 1ad, 1bd0 Lab sa e0 b. The identity Ls ¼ Ls is guaranteed by the fact that L(f ) ¼ f L( ) when f is a function. The same identity guarantees that the homomorphism L does not depend on the choice of the basis of sections. This is to say that any two choices give the same section of Hom(E; E0 ). N It is traditional to view   0 as a section of End(E) T*M rather than as a N section of Hom(E; E T*M); these bundles being canonically isomorphic. What was just said about the affine nature of the space of covariant derivatives has the following implication: Let  denote a covariant derivative on C1(M; E), and let s denote a section of E. Suppose that U is an open set in M and jU : EjU ! U  V is a bundle isomorphism. Write jU s as x ! (x, sU(x)) with sU : U ! V. Then jU(s) appears as the section x ! ðx;ðsÞU Þ where sU ¼ dsU þ aU sU N

and aU is some s-independent section of (U  EndV) T*MjU. Be forewarned that the assignment U ! aU does not define a section over M of N End(E) T*M. Here is why: Suppose that U0 is another open set with a bundle isomorphism jU0 : EjU0 ! U0  V that overlaps with U. Let gU0 ,U : U \ U0 ! End(V) denote the transition function. Thus, sU0 ¼ gU0 ,U sU. Meanwhile, s is a bonafide N section of E T*M and so (s)U0 ¼ gU0 ,U(s)U. This requires that 1 0 aU0 ¼ gU 0 ;U aU g1 U0 U  ðdgU ;U ÞgU0 U :

Conversely, a covariant derivative on C1(M; E) is defined by the following data: First, a locally finite, open cover L of M such that each U 2 L comes with a bundle isomorphism jU : EjU ! U  V. Second, for each U 2 L, a section of N aU : (U  EndV) T*MjU, this denoted by aU . The collection {aU}U2L define a covariant derivative if and only if the condition above holds for any pair U, U0 2 L that overlap.

11.3

Another construction of covariant derivatives

What follows describes a relatively straightforward construction of a covariant derivative on a given vector bundle E ! M with fiber Vn ¼ Rn or Cn. I assume here that M is compact, but with care, you can make this construction work in general. Let V denote either R or C as the case may be. This construction exploits the fact that the product bundle M  VN has one very simple covariant derivative that is defined as follows:

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Let x ! f ¼ (x, (f 1(x), . . . , f N(x))) denote a section of the bundle M  VN ! M. N Define the section df of (M  VN) T*M by x ! ðx; ðd f 1 jx ; : : :; d f N jx ÞÞ: This gives a covariant derivative. Now, suppose that N  1 and that E is a subbundle of M  VN. Introduce 2 Hom(M  VN; E) to denote the fiberwise orthogonal projection in VN onto E. Let s: M ! E now denote a section of E. Since E sits in M  RN, I can view s as a section of M  RN and so make sense of ds. This understood, then s ¼ ds N

is a section of E T*M, and the assignment s ! s is a covariant derivative on E. Now suppose we are given the bundle E ! M in isolation. Here is how to view E as a subbundle of some trivial bundle: Let E* ! M denote the bundle Hom(E; V). Recall that one can find some integer N  1 and a set {s1, . . . , sN} of sections of E* with the property that this set spans the fiber of E at each point in M. One way to do this is to use the open cover, L, of M that was introduced above, with its associated partition of unity {wU}U2L. Fix a basis {v1, . . . , vn} for V. Each U 2 L determines the set of n sections {s1U, . . . , snU} where skU ¼ jU1(wUvk). Here jU : EjU ! U  V is the associated isomorphism. Also, vk is viewed here as a section of U  V. Take the set {s1, . . . , sN} to be an ordering of the set U2L {s1U, . . . , snU}. Define a bundle homomorphism c: E ! M  VN by the following rule: Let x 2 M and v 2 Ejx. Then c(v) ¼ (x, s1jx v, . . . ., sNjx v). This puts E inside M  VN as a subbundle, and so we can use the definition of  above to define a covariant derivative for sections of E. Here is a relatively concrete example: View Sn as the unit sphere in Rnþ1. Now view TSn as the set of pairs {(x, v) : x, v 2 Rnþ1 with jxj ¼ 1 and xTv ¼ 0}. This identifies TSn with a subbundle in Sn  Rnþ1. Let s now denote a section of N TSn, thus a vector field on Sn. Define the section s of TSn T*Sn by the T rule s ¼ ds. Note that in this case jx ¼ i  xx where i is the identity (n þ 1)  (n þ 1) matrix. Since xTs(x) ¼ 0, another way to write this is s is s ¼ ds þ xdxT s. For instance, consider the following vector field: Let e 2 Rnþ1 denote a constant vector. Then x ! s(x) ¼ e  xxTe is a section of TSn. Its covariant derivative is (xxT  i)dxxTe. \

128

11.4

Principal bundles and connections

This part the chapter defines the notion of a connection on a principal bundle. This notion is of central importance in its own right. In any event, connections

11.4

Principal bundles and connections

are used momentarily to give an alternate and very useful definition of the covariant derivative. The discussion here has seven parts.

11.4.1

Part 1

Suppose that E is associated to some principal bundle p: P ! M. (We know that E is always associated to PGl(E), and to perhaps other principal bundles.) In particular, suppose that G is a Lie group, V ¼ Rm or Cm as the case may be, and r is a representation of G in Gl(V). Suppose further that E is given as P r V. Recall from Chapter 10.9 that the latter is the quotient of P  V by the relation (p, v) (pg1, r(g)v) when g 2 G. A section s of E appears in this context as a G-equivariant map from P to V. Said differently, a section s defines, and is conversely defined by a smooth map sP: P ! V that obeys sP(pg1) ¼ r(g)sP(p). Indeed, suppose first that one is given such a map. The corresponding section s associates to any given point x 2 M the equivalence class of (p, sP(p)) where p can be any point in Pjx. To see that this makes sense, remark that any other point in Pjx can be written as pg1 with g 2 G. Because sP(pg1) ¼ r(g)sP(p), the pair (pg1, sP(pg1)) defines the same equivalence class in P r V as does the original pair (p, sP(p)). To go in reverse, suppose that s is a section of P r V. By definition, s associates an equivalence class in P  V to every point in M. An equivalence class is an assignment to each p 2 P of a point v 2 V such that pg1 is assigned the point r(g) p. But this is just another way to say that such an assignment is a smooth map from P to V that obeys the required condition. This view from P of a section of P r V has the following useful generalization: Suppose that p: E0 ! M is a second vector bundle either real or complex, and perhaps unrelated to P. The examples that follow take E0 to be either TM, T*M or some exterior power ∧k T*M. In any event, a section over M of N (P r V) E0 appears upstairs on P as a suitably G-equivariant section over P of the tensor product of the product bundle P  V with the bundle p*E0 . Here, p*E0 is the pull-back of E0 via p; this bundle is defined in Chapter 5.1 as the subspace of pairs (p, e0 ) 2 P  E0 that share the same base point in M. Said N differently, a section of the tensor product bundle (P r V) E0 over M appears up on P as a G-equivariant, fiber-wise linear map from p*E0 to V. The notion of G-equivariance in this context is as in the preceding paragraph: If s denotes N E0 in question, then the corresponding map, sP, the section of (P r V) from p*E0 to V is such that sP(pg1, e0 ) ¼ r(g) sP(p, e0 ) for each g 2 G and pair (p, e0 ) 2 p*E0  P  E0 . Meanwhile, the notion of fiber-wise linear means the following: If r is in either R or C as the case may be, then sP(p, re0 ) ¼ r s(p, e0 ); and if (p, e10 ) and (p, e20 ) are points in p*E0 over the same point in P, then sP(p, e10 þ e20 ) ¼ sP(p, e10 ) þ sP(p, e20 ).

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11.4.2

Part 2

We already know how to take the derivative of a map from a manifold to a vector space. This Y is the manifold in question, and v the map from Y to the vector space, then the derivative is a vector of differential forms on Y; any given component is the exterior derivative of the corresponding component of v. In the case when Y ¼ P and v is the map sP that corresponds to a section, s, of P r V, then this vector of differential forms is denoted by, (sP) . As a vector of * differential forms, it defines a fiber-wise linear map from TP to V. Although this map is suitably G-equivariant, it does not by itself define a covariant derivative. Indeed, a covariant derivative appears upstairs on P as a G-equivariant, fiberwise linear map from p*TM to V. Said differently, a covariant derivative appears on P as a suitably G-equivariant section over P of the tensor product of p*TM with the product bundle P  V.

11.4.3

Part 3

To see what (sP)* is missing, it is important to keep in mind that TP has some special properties that arise from the fact that P is a principal bundle over M. In particular, there exists over P the sequence of vector bundle homomorphisms ð*Þ

0 ! kerðp Þ ! TP ! p*TM ! 0; *

where the notation is as follows: First, p : TP ! TM is the differential of the * projection map p and ker(p ) designates the subbundle in TP that is sent by p * * to the zero section in TM. This is to say that the vectors in the kernel p are * those that are tangent to the fibers of p. Thus, ker(p ) over Pjx is canonically * isomorphic to T(Pjx). The arrows in (*) are meant to indicate the following homomorphisms: That from ker(p*) to TP is the tautological inclusion as a subbundle. That from TP to p*TM sends a given vector v 2 TP to the pair (p(p), p*v) in p*TM  P  TM. What follows are three key observations about (*). Here is the first: The sequence in (*) is exact in the sense that the image of any one of the homomorphisms is the kernel of the homomorphism to its right. Here is the second: The action of G on P lifts to give an action of G on each of the bundles that appear in (*). To elaborate, suppose that g is a given element in G. Introduce for the moment mg : P ! P to denote the action of g 2 G that sends p to pg1. This action lifts to TP so as the push-forward map (mg)* as defined in Chapter 5.3. If v 2 TP is in the kernel of p*, then so is (mg)*v because p  mg ¼ p. Thus, (mg)* acts also on ker(p*). The action lift of mg to p*TM is defined by viewing the latter in the manner described above as a subset of P  TM. Viewed in this way, mg act so as to send a pair (p, v) 2 P  TM to the pair (pg1, v). Here, is the final observation: The homomorphisms in (*) are equivariant with respect to the lifts

11.4

Principal bundles and connections

just described of the G-action on P. Indeed, this follows automatically for the inclusion map from ker(p ) to TP, and it follows for the map from TP to p*TM * because p  mg ¼ p. A very much-related fact is that ker(p ) is canonically isomorphic to * the trivial bundle P  lie(G). This isomorphism is given by the map c: P  lie(G) ! ker(p ) that sends a pair (p, m) 2 P  lie(G) to the tangent * vector at t ¼ 0 of the path t ! p exp(t m). This map c is equivariant with respect to the action of G on lie(G) that sends g 2 G and m to gmg1. This is to say that the differential of the map mg : P ! P that sends p to pg1 act so that ðmg Þ* cðp; vÞ ¼ cðpg1; gvg1 Þ :

11.4.4

Part 4

A connection on the principal bundle P is neither more nor less than a G-equivariant splitting of the exact sequence (*). Thus, a connection, A, is by definition a linear map A : TP ! kerðp Þ * that equals the identity on the kernel of p and is equivariant with respect * to the action of G on P. This is to say that if g 2 G and v 2 TP, then (mg) (Av) ¼ A((mg)*v). * The isomorphism c: P  lie(G) ! ker(p ) is used to view a connection, A, * as a map A : TP ! lieðGÞ with the following properties:  A(c(p, m)) ¼ m.  If g 2 G and v 2 TP, then A(mg*v) ¼ gA(v)g1. This last view of A is quite useful. Viewed this way, a connection is a section of the tensor product of P’s cotangent bundle with the product bundle P  lie(G) with certain additional properties that concern the G-action on P and the sequence (*). The kernel of the homomorphism A is a subbundle in TP which is isomorphic via the right most arrow in (*) to p*TM. This subbundle ker(A) is often called the horizontal subbundle of TP and denoted by HA  TP. Meanwhile, the bundle ker(p ) is just as often called the vertical subbundle. * With regards to the notation in what follows, it is customary not to distinguish by notation the aforementioned two views of a connection, one as a bundle homomorphism from TP to ker(p ) and the other as a fiber-wise linear *

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map from TP to lie(G). In any event, the latter view is used primarily in what follows. Another notational quirk with regards to the second of these two views N uses T*P lie(G) to denote the tensor product of T*P with the product vector bundle P  lie(G).

11.4.5

Part 5

If A and A0 are connections on P, then aP ¼ A  A0 annihilates ker(p ). As a * consequence, it defines a fiber-wise linear, G-equivariant map from p*TM to lie(G). Given that aP is G-equivariant and linear on the fibers of p*TM, it corresponds (as described in Chapter 11.4.2 above) to a section over M of N (P ad lie(G)) T*M. Here, ad denotes the representation of G on its Lie algebra that has g 2 G sending any given matrix a 2 lie(G) to g a g1. As explained momentarily, the latter matrix is also in lie(G). This representation of G on lie(G) is called the adjoint representation. To say more about the adjoint representation, recall that the Lie algebra of G was defined to be the tangent space to the identity element i in G. Given that G is a matrix group, this vector space is a subspace of the relevant n 2 {1, 2, . . . } version of either Gl(n; R) or Gl(n; C). What follows explains how to see that g ag1 2 lie(G) if a 2 lie(G) and g 2 G. As explained in Chapter 8.7, the exponential map 1 a ! e a ¼ i þ a þ a2 þ : : : 2 maps lie(G) to G. Moreover, this map restricts to a ball about the origin in lie(G) so as to define a coordinate chart in G for a neighborhood of the origin. This implies in particular the following: Suppose that a is any given matrix. If t 2 (0, 1) is not too big, the matrix eta is in G if and only if a 2 lie(G). With the preceding as background, suppose that a 2 lie(G) and g 2 G. Fix some small t > 0. Use the 1 formula above to see that the matrix etgag ¼ g eta g1. Since all three matrices in this product are in G, the result is in G. If t is not too large, this implies that t g a g1 2 lie(G) and thus g a g1 2 lie(G). To continue the discussion about connections, suppose now that a is a section N T*M and that A is a connection on P. over M of the bundle (P ad lie(G)) Just as a section of P ad lie(G) can be viewed as a G-equivariant map from P to lie(G), so a can be viewed on P as a G-equivariant, fiber-wise linear map from p*TM to lie(G). Let aP denote the latter incarnation of a. Then A þ aP defines another connection on P. These last observations have the following corollary: If P has one connection, then it has infinitely many, and the space of smooth connections on P is an affine N space based on C1(M; (P ad lie(G)) T*M).

11.4

11.4.6

Principal bundles and connections

Part 6

What follows constructs a connection on P. As a preamble to the construction, suppose that U  M is an open set with a principal bundle isomorphism jU : PjU ! U  G. The product principal bundle U  G has a tautological connection 1-form, this defined as follows when G is a subgroup in Gl(n; R) or Gl(n; C). The connection 1-form at a point (x, g) 2 U  G is A0 ¼ g1 dg; this a matrix of 1-forms as it should be. Here, and in what follows, I view a connection on a principal G-bundle P as a lie(G)-valued 1-form P with the properties listed in Chapter 11.4.4. To see that A0 has the correct properties, fix a matrix m 2 lie(G) with lie(G) viewed as a subvector space in M(n; R) or M(n; C) as the case may be. Introduce the map t ! getm from R into G, this as described in Chapter 5.5. The pull-back of A0 by this map at t ¼ 0 is the 1-form on R given by (A0jg)(c(g, m)) dt. This is m dt, as it should be. Note also that A0 is suitably equivariant under the action of G on G by right translation. The lie(G)-valued 1-form A0 annihilates vectors that are tangent to the U factor of U  G. This understood, the corresponding horizontal subbundle HA0  TP are precisely the tangents to the U factor, this the factor TU in the L obvious splitting of T(U  G) ¼ T U TG. With this preamble in mind, now fix a locally finite cover U of M such that each set U 2 U comes with a principal G-bundle isomorphism jU : PjU ! U  G. Let {wU}U2U denote a subordinate partition of unity. Such a partition is constructed in Appendix 1.2. Set X w j *A : A¼ U2U U U 0 This lie(G)-valued 1-form on P is a connection. The verification is left as an exercise.

11.4.7

Part 7

This last part of the story gives another way to view a connection on P. To start, let U  M again denote an open set with a principal G-bundle isomorphism j: PjU ! U  G. Now let A denote a given connection on G. Then (j1 U )*A is a connection on the product bundle U  G. As such, it can be written as 1 1 ðj1 U Þ*A ¼ g dg þ g aU g

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where aU is a 1-form on U with values in lie(G). The horizontal space HA can be L described as follows: Identify T(U  G) ¼ T U TG and identify TG with lie(G) as done above. Then ðjU Þ* ðHA Þjðx;gÞ ¼ fðv; g1 aU ðvÞgÞ 2 TU

11.5

O

lieðGÞ: v 2 TUjx g:

Connections and covariant derivatives

Suppose that E ! M is a vector bundle which is given by P r V where r is a representation of G on V. Let A denote a connection on P. Then A defines a covariant derivative, A , on C1(M; E) as follows: View a section s of E ¼ P rV as a G-equivariant map, sP: P ! V. The corresponding covariant derivaN tive As will be viewed as a G-equivariant section, (As)p, of (P  V) p*TM. To define this section, introduce as in Chapter 11.4.2, the differential, (sP)*, of the map sP from P to V. This differential maps TP to V. Now, (As)(P) is N supposed to be a G-equivariant section over P of (P  V) p*T*M, which is to say a G-equivariant homomorphism from p*TM to V. To obtain such a thing, recall first that the connection’s horizontal subbundle, HA  TP is canonically isomorphic to p*TM. This understood, the restriction to HA of the homomorphism (sP)*: TP ! V defines a covariant derivative of s. The latter is, by definition, the covariant derivative As. Here is a reinterpretation of this definition. Let x 2 M and let v 2 TMjx. The covariant derivative As is supposed to assign to v a section of E. To obtain this section, pick any point p 2 Pjx. There is a unique horizontal vector vA 2 HAjp such that p vA ¼ v. The covariant derivative As sends v to the equivalence * class in Ejx ¼ (Pjx r V) of the pair (p, (sP)* vA). To see that this is well defined, I need to check that a different choice for p 2 Pjx gives the same equivalence class. That such is the case is a consequence of the fact that A and sP are suitably equivariant with respect to the G-action on Pjx. This is proved by unwinding all of the definitions—a task that I will leave to you. To see that this defines a covariant derivative, one need only check Leibnitz’s rule. To do so, suppose that f is a smooth function on M. Then (fs)P ¼ p*(f) sP N P P P and as a consequence, (fs )* ¼ p*(f) (s )* þ s p*(df). This implies that ðA ðfsÞÞp ¼ ðfA sA Þp þ sp

O

p* ðdfÞ:

The latter asserts Leibnitz’s rule. If A and A0 are connections on P, then the difference A0  A is supposed to N be a section over M of End(E) T*M. What follows identifies this section: Recall from Chapter 11.4.5 that the difference A0  A ¼ a can be viewed as a

11.6

Horizontal lifts

N

section over M of (P ad lie(G)) T*M. Meanwhile, the representation r induces a bundle homomorphism from P ad lie(G) to End(E) as follows: The bundle End(E) is the associated vector bundle P ad(r) Gl(V) where ad(r) here is the representation of G on GL(V) that has any given g 2 G act on m 2 Gl(V) as r(g)mr(g1). This understood, the differential of r at the identity in G defines a G-equivariant homomorphism r*: lie(G) ! Gl(V); this intertwines the representation ad on lie(G) with the representation ad(r). The latter homomorphism N T*M to gives a corresponding homomorphism from (P ad lie(G)) N End(E) T*M. This corresponding homomorphism is also denoted by r* because it is defined by ignoring the T*M factor. To elaborate, it is defined so N N as to send a given decomposable element m u to r*(m) u; and as any given element is a linear combination of decomposable elements, the latter rule is all that is needed. Granted all of the above, it then follows by unwinding the definitions that A0 − A ¼ r ðaÞ : *

Note that if P is the bundle PGl(E) of orthonormal frames in E, then P ad lie(G) is the bundle End(E). This and the preceding equation gives a 1–1 correspondence between covariant derivatives on E and connections on PGl(E).

11.6

Horizontal lifts

Let p: P ! M denote a principal G-bundle. Fix a connection, A on P. The connection gives a way to lift any given smooth map g: R ! M to a map gA: R ! P such that p  gA ¼ g. This is done as follows: Fix a point p0 in the fiber of P over the point g(0). The lift gA is defined by the following two conditions:  gA(0) ¼ p0  Let ]t denote the tangent vector to R. Then (gA)*]t 2 HA. Thus, the lift starts at p0 and its tangent vectors are everywhere horizontal as defined by the connection A. This is to say that A((gA) ]t) ¼ 0. * To see what this looks like, suppose that g crosses a given open set U  M where there is a bundle isomorphism jU: PjU ! U  G. Then jU(gA) is the map t ! (g(t), g(t)) where t ! g(t) makes the tangent vector to the path t ! (g(t), g(t)) 1 )*A ¼ g1dg þ g1aUg on U  G. horizontal as defined using the connection (jU L With T(U  G) identified as before with TU lie(G), this tangent vector is   d g ]t ; g1 g : * dt

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The latter is horizontal if and only if it has the form (v, g1aU(v)g) with v 2 TU. This understood, the map t ! g(t) must obey g1

d g ¼ g1 aU ðg ]t ÞjgðtÞ g * dt

which is to say that d g þ aU ðg ]t Þjg g ¼ 0: * dt If you recall the vector field theorem from Chapter 8.3 about integrating vector fields, you will remember that an equation of this sort has a unique solution given its starting value at some t ¼ t0. This being the case, horizontal lifts always exist.

11.7

An application to the classification of principal G-bundles up to isomorphism

The notion of a horizontal lift can be used to say something about the set of isomorphism classes of principal G-bundles over any given manifold. To this end, suppose that X is a smooth manifold and p: PX ! X is a principal G-bundle. Let M denote another smooth manifold. Theorem 11.2 Let f0 and f1 denote a pair of smooth maps from M to X. The principal G-bundles f0*PX and f1*PX are isomorphic if f0 is homotopic to f1. Proof of Theorem 11.2 There is an interval I  R containing [0, 1] and a smooth map, F, from I  M to X such that F(0, ) ¼ f0 and F(1, ) ¼ f1. Keeping this in mind, fix a connection, A, on the pull-back bundle F*P. Define a map cA: f0*P ! f1*P as follows: Let p 2 f0*P and let t ! gA,p(t) 2 F*P denote the horizontal lift of the path t ! (t, p(p)) 2 I  M that starts at p when t ¼ 0. Set cA(p) to be gA,p(1). This is an invertible, fiber preserving map. It also obeys cA(pg1) ¼ cA(p)g1 because of the equivariance properties of the connection. In particular, these guarantee that gA,pg1 ¼ gA,p g1 at all times t 2 I. The classification problem is completed with a second theorem which asserts the existence, for any given lie group G, of a universal classifying space, this a smooth manifold X such that any given principal G-bundle over any given manifold is the pull-back of a fixed principal bundle PX ! X. One can take dim(X) to be finite if a bound has been given a priori on the dimension of the manifolds under consideration. In any event, the result is that there is a 1–1 correspondence

11.8

Connections, covariant derivatives and pull-back bundles

{principal G-bundles on M up to isomorphism} , {homotopy classes of maps from M to X}. For example, the proposition in Example 10.14 asserts that the large m versions of real Grassmannian Gr(m; n) and the complex Grassmannian GrC (m; n) can serve as respective classifying spaces for principle O(n)- and U(n)-bundles over manifolds of a given dimension. Theorem 11.2 has the following as a corollary. Corollary 11.3 Let P ! M denote a principal G-bundle and let U denote a contractible, open set in M. Then there is an isomorphism j: PjU ! U  G. What follows is another corollary with regards to the case when M has dimesion 2. To set the stage, recall that Chapters 6.4, 6.5 and 6.11 described various complex, rank 1 vector bundles over surfaces in R3 that were pull-backs of a fixed bundle over S2 that was defined using the Pauli matrices. Corollary 11.4 Let M denote a compact, 2-dimensional manifold and let f0 and f1 denote homotopic maps from M to S2. Let p: E ! S2 denote a complex, rank 1 bundle. Then f0*E is isomorphic to f1*E if f0 is homotopic to f1.

11.8

Connections, covariant derivatives and pull-back bundles

Suppose that M and N are manifolds, that f: M ! N is a smooth map and that p: P ! N is a principal bundle with fiber a given group G. As explained in Chapter 10.2, the map f can be used to define pull-back principal bundle, pM: f*P ! M, this the subspace in M  P of points (x, p) with f(x) ¼ p(p). The projection, pM is induced by the obvious projection from M  P to its M factor. ^ f*P ! P that covers Meanwhile, there exists a canonical G-equivariant map f: f, this induced by the projection from M  P to its P factor. By the way, another way to view f*P is to consider P ¼ M  P as a principal G-bundle over the product M  N. Then M embeds in M  N as the graph of f, this the subset of pairs (x, f(x)). The bundle f*P is the restriction of the principal bundle P to the graph of f. Suppose that A is a connection on P; thus a lie(G)-valued 1-form on P that behaves in a specified manner when pulled back by the action of any given ^ element in G. Maps pull-back 1-forms, and the pull-back f*A is a lie(G)-valued 1-form on f*P. This 1-form satisfies all of the required conditions for a ^ is G-equivariant. You can check directly using the definiconnection because f tion of pull-back from Chapter 5.3; or you can see this by viewing P ¼ M  P as a principal G-bundle over M  N. Then A defines a connection on P, since it is

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a lie(G)-valued 1-form on P with all of the required properties. It just happens to be a 1-form with no dependence on the M factor in M  P. Granted all of ^ this, then f*A is the restriction of A (viewed as a connection on the bundle P ! M  N) to the graph of the map f. Now suppose that V is either Rn or Cn and r: G ! Gl(V) is a representation. The connection A on P gives the bundle E ¼ (P r V) a covariant derivative, . The pull-back bundle, f*E ¼ (f*P r V), then has the covariant derivative, ^ f*, this defined by the connection f*A. Here is another way to see this: View A again as a connection on the bundle P ¼ M  P over M  N. Let E denote the vector bundle M  E ! M  N. Of course, this is P r V. The restriction of E to the graph of f is the bundle f* E. The connection A, now viewed as living on P, defines a covariant derivative for E , and hence for the restriction of E to the graph of f. This covariant derivative is f*. What was just said about vector bundles can be said with no reference to connections. To elaborate, suppose that p: E ! N is a vector bundle, with no particular principal bundle in view. View M  E as a vector bundle, E, over M  N. As before, the pull-back bundle f*E is the restriction of E to the graph of f. If  is a covariant derivative for sections of E, then its restriction to sections of E over the graph of f defines the covariant derivative f* for sections of f*E.

Additional reading  Foundations of Differential Geometry, Volume 1, Shoshichi Kobayshi and Katsumi Nomizu, Wiley Interscience, 1996.  Differential Forms and Connections, R. W. R. Darling, Cambridge University Press, 1984.  Geometry, Topology and Physics, Mikio Nakahara, Taylor and Francis, 2003.  Lectures on Seiberg–Witten Invariants, John D. Moore, Springer, 2001.  Geometry of Four-Manifolds, S. K. Donaldson and P. B. Kronheimer, Oxford University Press, 1997.  Instantons and Four-Manifolds, Daniel S. Freed and Karen K. Uhlenbeck, Springer, 1990.

12

Covariant derivatives, connections and curvature

This chapter continues the discussion of connections and covariant derivatives with the introduction of the notion of the curvature of a covariant derivative or connection. This notion of curvature is defined in Chapter 12.4. The first three sections of the chapter constitute a digression of sorts to introduce two notions involving differentiation of differential forms and vector fields that require no auxiliary choices. Both are very important in their own right. In any event, they are used to define curvature when covariant derivatives reappear in the story.

12.1

Exterior derivative

This first part of the digression introduces the exterior derivative. As explained below, the exterior derivative is a natural extension to differential forms of the notion of the differential of a function. To start, suppose that f is a function on M. Chapter 3.9 describes how its differential, d f , can be viewed as a section of T*M. The association of any given function to its differential defines a linear map from C1(M) to C1(M; T*M). This understood, the exterior derivative extends this linear map so as to be a linear map from C1(M; ∧p T*M) to C1(M; ∧pþ1 T*M) for any given p 2 {0, . . . , n ¼ dim(M)}. Note in this regard that ∧p T*M is the 0-dimensional bundle M  {0} for p > n. This extension is also denoted by d. What follows defined d by induction on the degree, p, of the differential form. If f 2 C1(M) ¼ C1(M; ∧0 T*M), define df as in Chapter 3.9. (Recall that in a coordinate patch with coordinates (x1, . . . , xn), this P 1-form is df ¼ j ]x] j f dxj :Þ Granted the preceding, suppose next that d: C1(M; ∧p T*M) ! C1(M; ∧pþ1T*M) has been defined for all integers p ¼ 0, . . . , q  1 for 1  q  n. The following rules specify how d is to act on

C1(M; ∧q T*M):

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 If o 2 C1(M; ∧q T*M) and o ¼ da for some a 2 C1(M; ∧q1 T*M), then do ¼ 0.  If o ¼ f da for some a as before, and with f a smooth function, then do ¼ df ∧ da.  If o1 and o2 are two q-forms, then d(o1 þ o2) ¼ do1 þ do2. Note that this induction covers all cases. This is because any given point in M has a neighborhood on which ∧p T*M is spanned by a set of sections of the form dg1 ∧ . . . ∧ dgp for g1, . . . , gp chosen from a set of dim(M) functions. (To see this, take a coordinate chart centered on the point. Then the p-fold wedge products of the differentials of the coordinate functions span ∧p T*M on this coordinate neighborhood.) This definition must be checked for consistency. The point being that a given p-form o can be written as a sum of terms, each of the form gpþ1 dg1 ∧ . . . ∧ dgp, in many different ways. The consistency is equivalent to the following hallmark of partial derivatives: They commute: If f is a smooth function on Rn, then ] ] ] ] ]xj ]xk f ¼ ]xk ]xj f . To see why this is key, fix a coordinate chart so as to identify a neighborhood of given point in M with a neighborhood of the origin in Rn. Let f denote a function on M. Then d f appears in this coordinate chart as d f ¼ ∑j

] f dxj : ]xj

Now, if d(d f ) is to be zero, and also d(dxj) is to be zero, then the consistency of the definition requires that ∑k;j

] ] f dxk ∧ dxj ¼ 0: ]xk ]xj

The left-hand side above is indeed zero. The reason is that dxk ∧ dxj ¼ dxj ∧ dxk and so what is written here is the same as   1 ] ] ] ] ∑k;j f f dxk ∧ dxj ; 2 ]xk ]xj ]xj ]xk as can be seen by changing the order of summation. Meanwhile, what is written above is zero because partial derivatives in different directions commute. By the same token, suppose that a is a (p– 1)-form for p > 1. In this coordinate patch, a ¼ ∑1i1 < : : :

T*ðR  MÞ ¼ ð

>

p

>

142

T*MÞ

this is to say that a p-form a on R  M at any given point (t, x) can be written as dt ∧ ap1 þ ap where ap1 2 ∧p1T*Mjx and ap 2 ∧p T*Mjx. The form ap1 can be viewed as a time t 2 R dependent section of ∧p1 T*Mjx and the form ap is a time t 2 R dependent section of ∧p T*M. The exterior derivative da is a (p þ 1)-form on R  M and so it has corresponding components dt ∧ (da)p þ (da)pþ1. These are given by ðdaÞp ¼ d⊥ ap1 þ

] ap and ðdaÞpþ1 ¼ d⊥ ap ]t

where d⊥ is used here to denote the exterior derivative at any fixed t 2 R of a differential form on {t}  M.

12.3

Lie derivative

Now suppose that a ¼ c*o where c: [0, 1]  M ! N is the homotopy from f0 to f1. If o is closed, then dc*o ¼ c*do ¼ 0, so c*o is a closed form. This understood, the preceding formula for (dc*o)pþ1 says that the pull-back of o by c(t, ) is closed for any given t 2 [0, 1], and the formula for (dc*o)p can be integrated to see that ð1 * o þ d ðc*oÞ : f* o ¼ f p1 1 0 0

This proves that f1*o differs from f0*o by an exact form. This fact just stated has the following corollary: Proposition 12.1 Let U  M denote a contractible open set. This is to say that there exists a smooth map c: [0, 1]  U ! M such that c(1, ) is the identity map on U and c(0, ) maps U to a point p 2 M. Let p  1 and let o denote a closed p-form on M. Then there exists a (p  1)-form a on U such that da ¼ o. Proof of Proposition 12.1 It follows that ojU ¼ c(1, )*o and this differs from c(0, )*o by an exact form. Meanwhile, c(0, ) maps U to a point, and since ∧p T*{point} ¼ 0 for p > 0, so c(0, )*o is equal to zero. The proposition has as its corollary what is usually called the Poincare´ lemma: Corollary 12.2 (the Poincare´ lemma) Let U denote a contractible manifold, for instance Rn. Then any closed form p-form on U for p  1 is the exterior derivative of a (p 1)-form. This corollary can be used to prove the De Rham cohomology of a compact manifold is a finite dimensional vector space over R. See Chapter 9 in the book Differential Forms and Algebraic Topology by R. Bott and L. Tu (1982).

12.3

Lie derivative

The preceding discussion about the pull-back of a differential form via a homotopy gives a useful formula when the homotopy is a map c: R  M ! M that is obtained by moving the points in M some given time along the integral curves of a vector field. To elaborate, suppose that M is compact and that v is a vector field on M. The vector field theorem in Chapter 8.3 supplies a smooth map c: R  M ! M with the following properties:  c(0, x) ¼ x.  c*]t ¼ vjc .

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Let o now denote a p-form on M. Then c*o is a p-form on R  M. What can be said about the components (c*o)p1 and (c*o)p? The time derivative of (c*o)p at t ¼ 0 is said to be the Lie derivative of o with respect to v. It is denoted Lv o and is such that       ] ðc * oÞp jt¼0 ¼ Lv o ¼ do v; þ d o v; ]t where the notation is as follows: If u is a p-form on M and v is a vector field, then u(v, ) is the (p 1)-form that is obtained by the homomorphism N N N N N p1 TM ( p T*M) ! T*M that comes by writing TM ( p T*M) ¼ N N N p1 (TM N T*M) ( T*M) and then using the defining homomorphism from (TM T*M) to R. The form (c*o)p1 at t ¼ 0 is equal to o(v, ). The values of (c*o)p and (c*o)p1 at times t 6¼ 0 are obtained by using these t ¼ 0 formulae with o replaced its pull-back via the diffeomorphism c(t, ): M ! M. The Lie derivative acting on p-forms can be viewed as a derivation on an extension of the algebra of smooth functions. Recall that C1(M; R) is an algebra with addition and multiplication being point-wise addition and multiplication of functions. The derivations of this algebra are the vector fields, thus the sections of TM. Addition of forms and wedge product give L M ¼ 0pn C1(M; ∧p T*M) the structure of an algebra, albeit one in which multiplication is not commutative. This is an example of what is called a super algebra, which is a vector space with a Z / 2Z ¼ {0, 1} grading and a multiplication whereby ab ¼ (1) deg(a) deg(b) ba. The grading for M takes the p ¼ even forms to have grading degree zero and the p ¼ odd forms to have grading degree one. The point is that C1(M) is a subalgebra of M and the Lie derivative, v ! L v is an extension of the derivation that v defines on C1(M) to give a derivation of M. This is to say that L v (a ∧ b) ¼ L v a ∧ b þ a ∧ Lvb. You can check that this formula holds using the definition given above for Lv.

12.4

Curvature and covariant derivatives

Suppose that p: E ! M is a vector bundle with fiber V ¼ Rn or Cn. Also suppose that  is a covariant derivative on C1(M; E). As explained below, there is a nice N N extension of  that maps C1(M; E (∧pT*M)) to C1(M; E (∧pþ1T*M)) for each p 2 {0, 1, . . . , dim(M)}. Here, as before, ∧0 T*M is the vector bundle M  R, and ∧dim(M)þ1 T*M is the bundle with 0-dimensional fiber M  {0}. This extension is called the exterior covariant derivative, it is denoted as d, and it is defined by the following rules:

12.4

Curvature and covariant derivatives

 If o is a p-form and s is a section of E, then d(s o) ¼ s ∧ o þ s do. N  If w1 and w2 are sections of E ∧p T*M, then d(w1 þ w2) ¼ dw1 þ dw2. N

These rules are sufficient to define d on all sections of E ∧p T*M. Although d2 ¼ 0, this is generally not the case for d. It is the case, however, N that d2 defines a section of C1(M; End(E) ∧2 T*M). This section is denoted 2 w ¼ F ∧ w. The section F is by F and it is characterized by the fact that d called the curvature of the covariant derivative. It measures the extent to which the covariant derivatives in different directions fail to commute. More is said about this momentarily. 2 w can be written as F ∧ w with F an What follows directly explains why d End(E)-valued 2-form. Here is a first justification: Suppose that o is a smooth pform and s is a section of E. As d(s o) ¼ ds ∧ o þ s ∧ do, so d2 ðsoÞ ¼ d ðd s ∧ oÞ þ d ðs doÞ: Now, remember that if a is a 1-form, then d(a ∧ b) ¼ da ∧ b  a ∧ db. This being the case, the left-most term on the right-hand side above is 2 d s ∧ o  ds ∧ do. Meanwhile, the right-most term on the right-hand side is ds ∧ do þ s ∧ d2o. Since d2o is zero, the two terms on the right side above 2 add to d s ∧ o. In particular, if o ¼ f and thus a function, this says that d2 (fs) ¼ 2 fd s and so d2 commutes with multiplication by a function. This understood, apply Lemma 11.1 in Chapter 11.2 to see that d2 is given by the action of a N section of End(E) ∧2 T*M. This claim can be seen explicitly on a set U  M where there is a bundle isomorphism j: EjU ! U  V. Let d denote the covariant derivative that acts on a section x ! v(x) ¼ (x, v(x)) of U  V so as to send it to the section N x ! dvjx ¼ (x, dvjx) of the bundle (U  V) T*MjU. Then j(d(j1v)) acts to send the section v to the section given by x ! (x, dv þ a v) where a is a section N 2 (j1v)) is the section of End(E) T*MjU. This understood, j(d   x ! x; ðda þ a ∧ aÞv : This identifies the curvature 2-form F on U: jðF Þ ¼ da þ a ∧ a: What follows says more about the meaning of a ∧ a and, more generally, a ∧ b for End(E)-valued p and q forms a and b. To start, fix any given point in M and fix a basis {Za} for ∧p T*M and a basis {sb} for ∧q T*M at this point. Here, the index a runs from 1 to d!/p!(d  p)! where d ¼ dim(M). Likewise, the index b P P runs from 1 to d!/q!(d  q)!. Write a ¼ a aa Za and b ¼ b bb sb where {aa} and {bb} are in End(E). This done, then a ∧ b ¼∑a; b aa bb Za ∧sb ;

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where aa, bb 2 End(E) denotes the composition of the indicated endomorphisms. (This is just matrix multiplication.) In the case where a ¼ b and both are 1-forms, then what is written above is equivalent to the formula a ∧ a ¼∑1a< bd ½aa ; ab Za ∧Zb : In particular, a ∧ a ¼ 0 if and only if the various components of a pair-wise commute as endomorphisms of E. If U0 is a second open set with an isomorphism j0 : EjU0 ! U0  V, then 0 j d(j0 )1 can be written as d þ a0 . On U0 U, the End(E)-valued 1-form a0 can be obtained from a by the rule ¨

1 0 a0 ¼ gU0 ;U ag1 U0 ;U þ dgU ;U gU0 ;U

where gU0 ,U : U0 U ! Gl(V) is the transition function. This transformation rule is exactly what is needed so as to guarantee that ¨

146

da0 þ a0 ∧a0 ¼ gU0 ;U ðda þ a∧aÞ g1 U0 ;U : This last formula says that the identification of j(F) with the section x ! ðx; da þ a ∧ aÞ N

of (U  Gl(V) ∧2 T*MjU does indeed specify consistently over the whole of M an End(E)-valued 2-form.

12.5

An example

Let V ¼ RN or CN and suppose that E ! M is a subbundle of the product vector bundle M  V. Let P: M ! End(V) denote the map that sends any given point x 2 M to the orthogonal projection of V onto Ejx. Chapter 11.3 uses this setting to define a covariant derivative on sections of E. By way of a reminder, this is done as follows: View a section s: M ! E as a map to V whose value at each x 2 M happens to land in Ejx. Then define d s ¼ s ¼ Pds where d is the usual exterior derivative on maps from M to the fixed vector space V. Thus, ds is an N-dimensional vector at each point whose entries are 1-forms on M; any given entry is the differential of the function that defines the corresponding entry of s. To compute d2 , use the fact that Ps ¼ s to write ds as ds  (dP)s. Here and in what follows, dP is the N  N matrix of 1-forms that is obtained by differentiating the entries of P. Use this rewriting of ds with the fact that d2 ¼ 0 to see that

12.5

An example

d2 s ¼ PdðPÞ∧ds ¼ PdP∧dðP2 sÞ¼ðPdP∧dPPÞs: Here I have used two facts about P. First, PdPP ¼ 0 which can be seen by first differentiating the identity P2 ¼ P to get dPP þ PdP ¼ dP and then multiplying on the right by P. Second, Ps ¼ s. The preceding equation identifies the curvature 2-form: F ¼ PdP∧dPP: Note that what is written on the right-hand side here is bracketed fore and aft N by P and so it defines a bonafide section of End(E) ∧2 T*M. By way of a concrete example, suppose that M is an (n 1)-dimensional submanifold in Rn. As explained in Chapter 3.4, the tangent bundle TM can be viewed as the set of pairs (x, v) 2 M  Rn with n(x)Tv ¼ 0 where n(x) is the unit length normal vector to M at x. The corresponding covariant derivative is defined by the projection P ¼ i  n nT where i is the n  n identity matrix. Then PdP ¼ dn nT and dPP ¼ n dnT. This understood, it follows that F ¼ dn∧dnT: On the face of it, this defines at each point of M a skew-symmetric, n  n matrix of 1-forms whose (i, j) component is dni ∧ dnj where ni is the i’th component of the normal vector with the latter viewed as a map from M to Rn. The fact that jnj ¼ 1 at each point of M implies that this n  n matrix of 1-forms annihilates the vector n. Indeed, multiplying this matrix against n gives the Rn-valued 1-form P dn ∧ ð i dni ni Þ ¼ dn ∧ 12 dðjnj2 Þ ¼ 0. This being the case, what is written above does indeed define a 2-form on M with values in End(TM). For example, if M is the sphere Sn1 in Rn, then n(x) ¼ x. Near the north pole where xn ¼ 1, the collection of functions y ¼ (x1, . . . , xn1) can serve as coordinates for Sn1. This understood, then n(x(y)) has i’th coordinate yi if i < n and n’th coordinate (1  jyj2)1/2. Thus, dn is the vector of 1-forms whose i’th coordinate is dyi if i < n and whose n’th coordinate is (1  jyj2)1/2 yjdyj; where I am summing over repeated indices. Note that this vanishes at the north pole, as it should since n at the north pole has i’th coordinate zero for i < n and n’th coordinate 1. Meanwhile, dn ∧ dnT is a priori the n  n matrix of 2-forms whose i, j entry for i and j both less than n is dyi ∧ dyj, whose (i, n) entry for i < n is (1  jyj2)1/2 dyi ∧ yjdyj, whose (n, i) entry is minus what was just written, and whose (n, n) entry is zero. Note in particular that this matrix of 1-forms at the north pole has zero in its (i, n) and (n, i) entries for any i, and so defines (as it should) a 2-form with values in the endomorphisms of TSn1 at the north pole. This 2-form has (i, j) entry dyi∧dyj.

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12.6

Curvature and commutators

Let v and u denote vector fields on M. Viewed as derivations on C1(M), they have a commutator, denoted [v, u], which is also a derivation, and thus also a vector field on M. The derivation [v, u] sends a given function f to v(uf)  u(vf). Here, vf is shorthand for the action of v on the function f. It is also equal to the pairing between the 1-form df, a section of T*M, and the vector field v, a section of TM. This expression for [v, u] ¼ v(uf)  u(vf) obeys the required rule of a derivation: D(fg) ¼ Df g þ fDg. In a local coordinate chart where v ¼vi ðxÞ ]x] i and u¼ui ]x] i , then    ] ] i ] ½v; u ¼ vj j ðui Þ  uj v : j ]x ]x ]xi Here, I am summing over repeated indices. The commutator and exterior derivative are related in the following way: Let a denote a 1-form on M. Then            da v; u ¼ a ½v; u þ v a u  u a v : N This formula views a section of ∧p T*M as a homomorphism from pTM to R which happens to be completely antisymmetric with respect to permutations of the factors in the p-fold tensor product. For example, a(v) denotes the function on M that is obtained by evaluation of a on the vector field v and (da)(v, u) that N obtained by evaluating da on v u. The preceding formula has the following application: Let E ! M denote a vector bundle and let  denote a covariant derivative for C1(M; E). Let v and u denote vector fields on M, and let v and U denote the respective directional derivations on C1(M; E) that are obtained from v and u using . This is to say that if s is a section of E, then vs is the section of E that is obtained by pairing N the section s of C1(M; E T*M) with v. The commutator of these directional covariant derivatives is given by ½v ; u þ ½v; u ¼ F ðv; uÞ: This gives another way to define F. Here, F is viewed as an endomorphism N from 2T*M to P ad lie(G).

12.7

Connections and curvature

Suppose that P ! M is a principal G-bundle and A is a connection on P. The connection defines a covariant derivative on sections of any vector bundle that

12.7

Connections and curvature

is associated to P via a representation of G into a general linear group. Each such covariant derivative has an associated curvature 2-form. All of these 2-forms are derived from a single section, FA, of the bundle N 2 (P ad lie(G)) ∧ T*M. The latter section is called the curvature 2-form or just curvature of the connection. What follows defines FA and explains how it gives the curvatures of all of the various covariant derivatives. A first way to proceed invokes what has been said about P over open sets in M where P is isomorphic to the product principal G-bundle. To start, suppose that U  M is an open set and that there is a given bundle isomorphism j: PjU ! U  G. As explained in Chapter 11.4.7, the lie(G)-valued 1-form (j1)*A on U  G at a given point (x, g) 2 U  G can be written as ðj1 Þ*A ¼ g1 dg þ g1 aU g; where aU is a 1-form on U with values in lie(G). Here, as before, I am assuming that G is a subgroup of Gl(n; V) with V ¼ R or C as the case may be. Thus, lie(G) is a subvector space of n  n matrices and dg is an n  n matrix whose entries are the coordinate 1-forms on Gl(n; V). Now, let V denote a vector space, and let r: G ! Gl(V) denote a representation of G. As explained in Chapter 11.5, the connection A induces a covariant derivative, A, on the sections of P r V. Meanwhile, the isomorphism jU supplies a bundle isomorphism (also denoted by j) from (P r V)jU ! U  V. Let s denote a section of P r V. Then j(s) sends x 2 U to a section of the form (x, sU(x)), where sU is a smooth map to the vector space V. In addition, j(As) N is the section over U of (U  V) T*MjU given by x ! ðx; dsU þ r* ðaU ÞsU Þ; i. It follows where r*: lie(G) ! End(V) is the differential of r at the identity, N ∧2 T*M has from this equation that the section j(FA) of (U  End(V)) the form x ! ðx; r* ðdaU þ aU ∧aU ÞÞ: Note in particular how this expression for the curvature of A is using the lie(G)-valued 1-form aU that defines the connection A. With the preceding in mind, introduce the lie(G)-valued 2-form ðFA ÞU ¼ daU þ aU ∧aU : N

The assignment of x 2 U to (x, (FA)Ujx) is a section over U of (U  lie(G)) ∧2 T*M. The curvature 2-form FA can be defined by declaring that this section be its image under j. Note in this regard that if U0  M is another open set with an isomorphism j0 : PjU ! U0  G and such that U U0 6¼ , then aU0 and aU are related by the rule

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aU0 ¼ gU0 ;U aU ðgU0 ;U Þ1  dgU0 ;U ðgU0 ;U Þ1 where gU0 ,U ¼ j0  j1: U U0 ! G is the transition function. This transforma0 tion rule guarantees that daU0 þ aU0 ∧ aU0 ¼ gU0 ,U(daU þ aU ∧ aU)gU1 ,U. This transformation rule asserts the following: The assignment of (FA)U ¼ daU þ aU ∧ aU to any given open set U ! M with isomorphism j: PjU ! U N  G defines the image via j of a bonafide section of (P ad lie(G)) ∧2 T*M. This section is the curvature 2-form FA. Granted this definition of FA, what was said two paragraphs back tells us that the curvature 2-form FA of the covariant derivative A as defined on sections of P r V is equal to r* (FA). ¨

150

12.8

The horizontal subbundle revisited

What follows descibes a definition of FA that does not introduce isomorphisms of P with the product bundle over open sets in M. To start, go up to P and reintroduce the horizontal subbundle HA  TP. Suppose that vA and uA are a pair of sections of HA. These are, of course, sections of TP also. This understood, one can consider their commutator [vA, uA]. This may or may not be a section of HA. If it lies in HA, then its image via A in lie(G) is zero. Let v, u denote vector fields on M and let vA, uA denote horizontal lifts as sections over P of HA as defined in Chapter 11.6. This is to say that vAjp2P is the unique lift of vjp(p) to TPjp that lies in the horizontal subspace HAjp. Then the assignment of the ordered pair (v, u) ! A([vA, uA]): P ! lie(G) has the following properties:  A([vA, uA])jpg1 ¼ g A([uA, vA]) g1 .  If f, f0 2 C1(M), then A([(fv)A, (f0 u)A]) ¼ f f0 A([vA, uA]).  A([vA, uA]) ¼ A([uA, vA]). The first properties assert the fact that the assignment of the ordered pair (v, u) to A([vA, uA]) defines a section over M of the associated vector bundle P ad lie(G). The second property implies that the assignment of the ordered pair (v, u) to A([vA, uA]) does not involve the derivatives of v and u. This the case, it can be viewed as the result of evaluating a section over M of N N2 ( T*M) on the pair (v, u). The third property asserts that (P ad lie(G)) N N2 this section of (P ad lie(G)) ( T*M) changes sign when v and u are N interchanged and so defines a section of (P ad lie(G)) ∧2 T*M. This section is FA. N To see that this section of (P ad lie(G)) ∧2 T*M is the same as that defined previously, one need only consider this just completed definition over an open set U  M where there is an isomorphism j: PjU ! U  G. As explained in

Additional reading

Chapter 11.6, the push-forward of the horizontal lift, vA, of a vector field v is the vector field on U  G whose components at any given (x, g) with respect to the L identification T(U  G) ¼ TU lie(G) are (v, gaU(v)g1). This understood, it is an exercise with the definition of the Lie bracket to verify that the two definitions of FA agree on any given pair of vectors (v, u) 2 TU.

Additional reading  Introduction to Smooth Manifolds, John M. Lee, Springer, 2002.  An Introduction to Manifolds, Loring W. Tu, Springer, 2007.  Geometry of Differential Forms, Shigeyuki Morita and Teruko Nagase, American Mathematical Society, 2001.  The Geometry of Physics, Theodore Frankel, Cambridge University Press, 2003.  Geometry, Topology and Physics, Mikio Nakahara, Taylor and Francis, 2003.  Manifolds and Differential Geometry, Jeffrey M. Lee, American Mathematical Society, 2009.

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13

Flat connections and holonomy

Let G be a Lie group and p: P ! M a principal G-bundle. As explained in Chapter 11.4.5, there are infinitely many connections on P. This being the case, it has proved profitable to look for connections that satisfy certain auxiliary conditions. These have been, for the most part, constraints on the curvature 2-form. In particular, connections obeying such constraints have played a central role in many of the recent and most exciting research in differential geometry and in applications of differential geometry to other fields. The simplest of the curvature constraints asks for connections with vanishing curvature 2-form. This sort of connection is said to be flat. This chapter studies the flat connections. Various chapters to come discuss some of the other useful curvature constraints. The story told here introduces some auxiliary notions that are important in their own right; and many are used in subsequent chapters. What follows is a list of these auxiliary notions in their order of appearance:    

foliations the group of automorphisms of a principal bundle—the gauge group the fundamental group of a manifold covering spaces.

The last two of these notions are used at the end of this chapter to describe the set of all flat connections on all of the principal G-bundles over a given manifold M.

13.1

Flat connections

A connection on a principal bundle is said to be flat when its curvature 2-form is identically zero. The simplest example is the connection on the product principal bundle P ¼ M G that is defined by the identification

13.2

Flat connections on bundles over the circle

L

T(M  G)j(x, g) ¼ TMjx TGjg. This connection is the lie(G)-valued 1-form given by the formula Aj(x, g) ¼ g1dg. The latter 1-form is pulled back from the projection M  G ! G. There are other flat connections. For example, let h: M ! G denote any given smooth map. This defines a principal bundle isomorphism j: M  G ! M  G that sends any given pair (x, g) to (x, h(x)g). The pull-back, j*A, is the connection given by ðj*AÞjðx; gÞ ¼ g1 dg þ ðh1 dhÞjx ; here h1dh is shorthand for the lie(G)-valued section of T*M that sends a vector v 2 TMjx to the vector in lie(G) that corresponds via the isomorphism TG ¼ G  lie(G) to the push-forward h*v. To be even more explicit, assume that G is a subgroup of Gl(n; V) for some n and for V ¼ R or C. Then h is just a map from M to the vector space of n  n matrices; and as such, any given component is a function on M. Granted that such is the case, dh is the n  n matrix valued 1-form whose i, j component is the differential of the function that defines the i, j component of h. A calculation finds that the curvature of j*A is also zero. Indeed, the latter is the lie(G)-valued 1-form dðh1 dhÞ þ h1 dh ∧ h1 dh which is zero because dh1 ¼ h1dh h1. There may be other sorts of flat connections on M  G. The case where M ¼ S1 supplies an instructive example.

13.2

Flat connections on bundles over the circle

The manifold M is the circle, S1. Assume that G is connected. Let p: P ! S1 denote a principal G-bundle over S1. Every connection on P is flat as all 2-forms on S1 are zero. This being the case, the following is the truly interesting question: Suppose that A is any connection on P: Is there an isomorphism j: P ! S1  G such that ðj AÞðx; gÞ ¼ g1 dg? * To answer this question, note first that there is, in all cases, a principal G-bundle isomorphism j: P ! S1  G. To find one, it proves convenient to write S1 as R / 2pZ. Fix an open interval U  S1 containing 0 with a principal G-bundle isomorphism, j, from PjU to U  G. Also, fix a connection, A, on P and a point p 2 Pj0. Let gA,p : [0, 2p] ! P denote the horizontal lift of the path given by the projection map from [0, 2p] 2 R to S1. This lifted path is defined in Chapter 11.6. As Pj2p ¼ Pj0, there exists an element hA,p 2 G such that gA,p(2p) hA,p ¼ p. As G is path connected, there is a map, h: [0, 2p] ! G such

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that h near 0 is equal to i, and h(2p) ¼ hA,p. This understood, the map s: [0, 2p] ! P given by t ! gA,p(t)h(t) is such that s(0) ¼ s(2p) ¼ p. Thus, s defines a continuous section of P. As it turns out, the section s can be chosen so that the resulting section is smooth. This follows from what is said in Appendix 13.1 of this chapter on the topic of smoothing continuous maps. Note in this regard that the smoothing discussion from Appendix 13.1 can be applied directly after first fixing an interval I  S1 that contains 0 with a principal bundle isomorphism j: PjI ! I  G. Having done this, write the j-image of the original section of P as a map t ! (t, g(t)) with g a continuous map from I to G that is smooth except at 0. It can be approximated by a smooth map to obtain the desired smooth section of P. Granted that s is smooth, then an isomorphism j: S1  G ! P is given by the rule that sends (t, g) 2 S1  G to s (t) g 2 Pjt. Note that j(t, g g0 ) ¼ j(t, g)g0 for any given g0 2 G, so this does indeed define a principal G-bundle isomorphism. To return to the question posed in italics above, choose a principal G-bundle isomorphism P ! S1  G so as to view the given connection A on P as a connection on the product bundle S1  G. This done, then the connection can be written as the lie(G)-valued 1-form g1dg þ g1n g dt where n is a smooth map from S1 to lie(G). Thus, a ¼ n dt is a lie(G)-valued 1-form on S1. With P written as S1  G, the question in italics above is equivalent to the following: Let a denote a given lieðGÞ-valued 1- form on S1 : Is there a principal G-bundle isomorphism j: S1  G ! S1  G such that j*ðg1 dg þ g1 n g dtÞ ¼ g1 dg: To answer this question, introduce again the horizontal lift, gA,(0,i), of the tautological path [0, 2p]  S1 that starts at the point (0, i) 2 S1  G. Write gA,(0,i)(2p) as (0, hA,(0,i)) with hA,(0,i) 2 G. The desired isomorphism exists if and only if the element hA;ð0; iÞ 2 G as defined above is the identity element i: To see why this is, note first that hg–1dg,(0,i) ¼ i. Note second that any given bundle automorphism, j, of S1  G can be written as (t, g) ! (t, u(t) g) where u: S1 ! G. Indeed, u(t) is such that j(t, i) is (t, u(t)). This understood, it follows that hj*A,(1,i) ¼ u(0) hA,(0,i) u(0)1. Thus, j*A ¼ g1dg if and only if i is conjugate in G to hA,(0,i); and as i is the only element conjugate to i, so hA,(0,i) ¼ i. What follows describes how to find a connection with any given value of hA,(0,i): Suppose that the desired value is h 2 G. Fix a map t ! k(t) from [0, 2p] that sends a neighborhood of 0 to i and a neighborhood of 2p to h. Let mðtÞ ¼ ðdtd kÞk1 . Then the connection A ¼ g1dg  g1mg dt has hA,(0,i) ¼ h. To see why, recall from Chapter 11.6 that the horizontal lift of the path t ! [0, t) 2 R / 2pZ is the path in S1  G that sends any given t 2 R / (2pZ) to (t, g(t)) where t ! g(t) obeys the equation

13.3

Foliations

d g ¼ ag: dt In the case at hand, this equation asserts that that starts at i when t ¼ 0 is g(t) ¼ k(t).

13.3

d dt g

d ¼ ðdt kÞk1 g and the solution

Foliations

Let M denote a given manifold and P ! M a given principal G-bundle. Suppose that A is a flat connection on P. Let vA and uA denote any two sections of the horizontal bundle HA  TP as defined in Chapter 11.4.4. Since FA ¼ 0, it follows that their commutator is also a section of HA. This is to say that HA is involutive. The notion of an involutive subbundle occurs in a wider context of great import in its own right. To elaborate, let X denote any given manifold, and let H denote a subbundle of TX. The subbundle H is said to be involutive when the commutator of any two sections of H is also a section of H. The following theorem gives some indication of the significance of this notion. The proof of this theorem is given in Appendix 13.2 to this chapter. Theorem 13.1 (the Frobenius theorem) Suppose that X has dimension n and that H  TX is an involutive subbundle of dimension m  n. Then any point p 2 X has a neighborhood, U, with a coordinate chart j: U ! Rn ¼ Rm  Rnm such that j*H consists of the tangent vectors to the Rm factor in Rm  Rn. One says in this case that H is tangent to an m-dimensional foliation of X. The image via j1 of Rm  {x} for any x 2 Rnm is part of what is called a leaf of the foliation. Each leaf is a manifold in its own right, and any given point in this leaf has a neighborhood that is a submanifold, this being the image via j1 of Rm  {x} for some x 2 Rnm. However, the leaf itself need not be a submanifold unless it is compact. An example to ponder is the 1-dimensional foliation of the torus S1  S1 that is defined by the subbundle H spanned by the vector field 1 ] ] ] ]t1 þ a ]t2 where a is an irrational number. Here, I write S as R / 2pZ and use ]t to denote the corresponding Euclidean vector field. As noted at the outset, the horizontal subbundle HA  TP is involutive if FA ¼ 0. It follows from the definition of FA that the converse is also true: If HA is involutive, then FA ¼ 0. Thus, HA is tangent to the leaves of a foliation of P. This is called the horizontal foliation. The action of G on P maps leaves to leaves since HA is a G-equivariant subbundle. Furthermore, the cotangent space to each leaf is isomorphic to the pull-back via p of T*M.

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What follows is a converse of sorts to what was just said. Let F denote a foliation of P with two properties: First, the G-action maps leaves to leaves. Second, the cotangent bundle of each leaf is isomorphic to the pull-back via p of T*M. Then there is a flat connection A on P whose horizontal space is the tangent space to the leaves of the foliation. To elaborate, let H  TP denote the subbundle of vectors that are tangent to the leaves of F . This will be the horizontal subbundle of the desired connection. To obtain the connection, note that the equation labeled (*) in Chapter 11.4.3 implies that there is a unique homomorphism from TP to ker(p*) which has the following two properties: It annihilates vectors in the subbundle H and it acts as the identity on vectors in ker(p*). This map from TP to ker(p*) is the connection A, here viewed as in Chapter 11.4.4.

13.4

Automorphisms of a principal bundle

Suppose that P is a principal bundle over a given manifold. An automorphism of P is an isomorphism f: P ! P. The set of automorphisms is denoted by Aut(P). This is a group. It is also called the gauge group, which is a term that comes from our physics brethren. This group is, in fact, the group of sections of the associated fiber bundle P Ad G; this being (P  G)/ where the equivalence relation identifies any given pair (p, g) 2 P  G with (pg1, g g g1) when g 2 G. A section of this bundle is neither more nor less than a G-equivariant map from P to G. This is to say that a section of P Ad G can be viewed as a map u: P ! G with the property that u(pg1) ¼ g u(p)g1. Such a map defines the automorphism fu that sends any given p 2 P to p u(p). Note that fu(pg1) ¼ fu(p)g1 is guaranteed by the equivariance condition that is imposed on the map u. To see that all automorphisms arise in this way, suppose that f is any given automorphism. As f preserves the base point of p, the point f(p) can be written as pu(p) for some map u: P ! G. The fact that f(pg1) ¼ f(p)g1 now demands that u(pg1) ¼ g u(p) g1. The group Aut(P) is, in some well-defined sense, infinite dimensional. To see why it is so big, remark that its ‘Lie algebra’ is the space of sections of the vector bundle C1(M; P ad lie(G)). The point being that if w is a section of C1(M; P ad lie(G)), then exp(w) is a section of P Ad G. Here, I view w as a map from P to lie(G) so as to view exp(w) as the map that sends any given p 2 P to exp(w(p)) 2 G. It is a direct consequence of Chapter 5.4’s definition of the exponential map as expðwÞ ¼ 1 þ w þ

1 2 1 3 w þ w þ 2! 3!

13.5

The fundamental group

that its application in this context defines a map from P ad lie(G) to P Ad G that fixes the base point in M. Connections A and A0 on P are said to be gauge equivalent when there is an element f 2 Aut(P) such that A0 ¼ f*A. An important point here is that the respective curvature 2-forms of A and f*A are related as follows: Let h 2 C1(M; P Ad G) denote the equivariant map that defines f. Then ðFj *AÞjx ¼ f* ðFA jx Þ ¼ ðh1 FA hÞjx : N

To elaborate, remember that FA is a section of (P ad lie(G)) ∧2 T*M. Meanwhile, an automorphism, f, of P, induces an automorphism of all bundles that are associated to P, in particular, the bundle P ad lie(G). The automorphism here is just pointwise conjugation by the element h 2 C1(P Ad G) that defines f. Indeed, if V is a vector space and r: G ! Gl(V) is a group homomorphism, the automorphism f acts on the bundle P r V ¼ (P  V)/ as follows: Let h: P ! G denote the equivariant map that defines f. Then f sends the equivalence class of any given pair (p, v) 2 P  V to that of (p h(p)1, v); and this is the same equivalence class as (p, r(h(p)1) v). Let U  M denote an open set where there is an isomorphism jU : PjU ! U  G. Suppose that f is an automorphism of P. Then jU 8 f 8 j1 U is an automorphism of U  G. As such, it acts so as to send any given pair (x, g) to (x, z(x, g)). Here, the map z cannot be just any map from U  G to G. It must be equivariant with respect to the G-action and this implies that z(x, g) ¼ hU(x) g where hU(x) ¼ z(x, i) is a map from U to G. Note that if U0 is another open set with an isomorphism jU0 : PjU0 ! U0  G, then there is a corresponding map hU0 : U0 ! G and the pair hU and hU0 are determined one from the other on U \ U0 by the transition 0 function gU0 ,U via the rule hU0 ¼ gU0 ,U hU gU1 ,U. Let A denote a connection on P. The action of f on a connection A can be seen on PjU as follows: Write (jU1)*A ¼ g1dg þ g1aUg where aU is a lie(G)-valued 1-form on U. Then 1 1 1 1 ðj1 U Þ*ðj*AÞ ¼ g dg þ g ðhU dhU þ hU aU hU Þg:

As can be seen from the formula for Ff*A ¼ f*(FA), the group Aut(P) preserves the set of flat connections on P.

13.5

The fundamental group

The list of all flat connections on a given manifold M refers to the fundamental group of the manifold M. This is a discrete group that is determined by the underlying topological manifold. It is usually denoted by p1(M). To define this

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group, fix a point x 2 M. Let Lx denote the space of continuous maps from [0, 1] into M such that g(0) ¼ g(1) ¼ x. Define an equivalence relation, , on Lx by declaring that g0 g1 if there is a smooth homotopy, f , from [0, 1]  [0, 1] into M such that f (0, ) ¼ g0, f (1, ) ¼ g1, and such that f ( , 0) ¼ f ( , 1) ¼ x. This can be viewed as a path in the space Lx from g0 to g1. The set of equivalence classes is the set of points in p1(M). This set has the structure of a group, with constant map to x playing the role of the identity element. The inverse of an element represented by a loop g is the loop that traverses g in the reverse direction; this is the loop t ! g(1  t). Group multiplication is defined as follows: Let z0 and z1 denote elements in p1(M). Choose respective loops g0 and g1 to represent these elements. Then z1z0 is represented by the loop that traverses first g0 at twice normal speed so as to arrive back at x at time t ¼ 12, and it then traverses g1 at twice normal speed to arrive back at x at t ¼ 1. Note that p1(M) does depend on the choice of the point x. Even so, a different choice produces an isomorphic group. The manifold M is said to be simply connected when p1(M) ¼ {1}. For example, Rn is simply connected. So is Sn for n > 1. However p1(S1) ¼ Z. Let q > 0 and let M denote a compact surface of genus q. If q ¼ 1, then M is the L torus and p1(M) ¼ Z Z. If q > 1, then p1(M) is the quotient of the free group on 2q generators {z1, . . . , z2q} by the relation 1 : : : 1 1 ðz2q1 z2q z2q1 z2q Þ ¼ i: ðz1 z2 z1 1 z2 Þ

By way of reminder, a free group on generators {z1, . . . , zm} consists of the set of finite, ordered strings of the form y1y2. . . . where each yi is from the set k2Z{0} k {z1k, . . . , zm } where it is understood that if yi is a power of zk then neither yi1 nor yiþ1 is a power of zk. A homomorphism from p1(M) into a Lie group G is a map r: p1(M) ! G such that r(z1z0) ¼ r(z1)r(z0). For example, a homomorphism from Z (¼ p1(S1)) into G consists of the specification of a single element g 2 G. With g specified, r L sends any given integer k to gk. A homomorphism from p1(S1  S1) ¼ Z Z into G requires the specification of elements g1, g2 2 G that commute in the 1 sense that g1g2g1 1 g2 ¼ i. Given such elements, r sends the ordered pair of 1 integers (n, m) to g1n g2m. The condition g1g2g1 1 g2 ¼ i is what is needed to guarantee that r is a homomorphism. Let M denote a compact surface of genus q > 1. Then a homomorphism of p1(M) into G is given by the specification 1 of 2q elements {g1, . . . , g2q} 2 G subject to the condition that (g1g2g1 1 g2 ) . . . 1 1 (g2q1g2qg2q1g2q ) ¼ i. The set of homomorphisms from p1(M) to G is denoted by Hom(p1(M); G). Homomorphism r and r0 from p1(M) to G are said to be conjugate when there exists a fixed element g 2 G such that r0 ( ) ¼ g r( )g1. The notion of conjugacy defines an equivalence relation on the set Hom(p1(M); G). The set of equivalence classes is often written as Hom(p1(M); G)/G. \

158

13.7

13.6

The universal covering space

The flat connections on bundles over M

The theorem below describes all of the flat connections on principal G-bundles over a given manifold M. The theorem refers to the set FM,G defined as follows: This is the set of equivalence classes of pairs (P, A) where P ! M is a principal G-bundle and A is a flat connection on P; and where the equivalence relation (P, A) (P0, A0 ) when there exists an isomorphism j: P ! P0 such that j*A0 ¼ A. Theorem 13.2 (classification theorem for flat connections) The set FM,G is in 1–1 correspondence with the set Hom(p1(M); G)/G. This theorem is proved shortly. The next two sections of this chapter introduce two important notions that enter the proof of this classification theorem.

13.7

The universal covering space

Let M denote a given smooth manifold. A covering space is a smooth manifold, ˜ , with a map p: M ˜ ! M with the following property: Any given point p 2 M M has a neighborhood, U, such that p’s restriction to each component of p1(U) is a diffeomorphism onto U. The basic facts about covering spaces can be found in most any book on algebraic topology, for example the book Algebraic Topology: A First Course, by Greenberg and Harper (1981). What follows summarizes the facts that are used in the proof of the classification theorem (Theorem 13.2). The universal covering space of M is a simpy connected covering space. The choice of a base point x 2 M and a base point x˜ 2 p1(x) gives a canonical identification between p1(x) and the fundamental group of M. Note that any two universal covering spaces of the same manifold M are diffeomorphic via a diffeomorphism that intertwines the respective maps to M. Moreover, such a diffeomorphism is canonical given that it maps the base point in one to the base point in the other. Every manifold has a universal covering space. What follows explains how to construct one: Let P x denote the set of maps from [0, 1] to M that send 0 to x. ˜ is the quotient of P x by the equivalence relation that identifies Then M g0 with g1 when g0(1) ¼ g1(1) and the two maps are homotopic via a map f : [0, 1] ! P x such that f (s, 1) ¼ g0(1) ¼ g1(1) for all s. ˜ and M ¼ M ˜ /p1(M). The group p1(M) acts as a group of diffeomorphisms of M The action of a class z 2 p1(M) can be defined as follows: Fix a representative loop n 2 P x for the class z. Let n1 denote the loop that is obtained from n by

159

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traversing the latter in the reverse direction. Given g 2 P x, construct the concatenation g n1 2 P x. Then z sends the equivalence class of g to that of g n1. This diffeomorphism is denoted by cz. (Note that n1 must be used instead of n so as to have czz0 ¼ cz 8 cz0 .)

13.8

Holonomy and curvature

Fix an orientation on S1 and a base point 0 2 S1; then fix an orientation preserving identification between S1 with R / 2pZ so that 0 2 R is mapped to the base point. Let p: P ! M denote a principal G-bundle with connection A. Meanwhile, suppose that g: S1 ! M is a smooth map. Fix a point p 2 Pjg(0). As explained in Chapter 11.6, the loop g has canonical lift to a path gA,p : [0, 2p] ! P which is characterized by the fact that gA,p(0) ¼ p and ðgA;p Þ* ]t] is horizontal. In general, there is no need for gA,p(2p) to equal p. However, as gA,p(2p) 2 Pjg(0), one can write gA(2p) ¼ p hA,g(p) where hA,g is a map from Pjg(0) ! G that obeys hA,g(pg1) ¼ g hA,g (p) g1. This map is called the holonomy of the connection A around the based, oriented loop g. The holonomy hA,g depends on the chosen base point 0 2 g, and it depends on the orientation of g; but it is otherwise independent of the chosen identification of S1 as R / 2pZ. The assignment of a based, oriented loop in M to the holonomy has four key properties. What follows are the first three. Property 13.3 Let *: S1 ! M map all of S1 to a point in M. Then hA,*(p) ¼ i. Property 13.4 Let g1: S1 ! M denote the traverse of g in the direction opposite to that defined by its given orientation. Then hA,g1(p) ¼ (hA,g(p))1. Property 13.5 Let m: S1 ! M and n: S1 ! M denote smooth, oriented maps such that m(0) ¼ n(0). Define the concatenation n m by first traversing m and then traversing n. The holonomy of the concatentated loop is given by hA,n m(p) ¼ hA,n(p) hA,m(p). All of these properties follow directly from what was said in Chapter 11.6 about horizontal lifts of paths in M. The details are left for you to work out. The fourth property is a formula for how hA,g changes when g is changed. What follows is a digression to set the stage for this formula. To start this digression, fix a smooth map g: S1 ! M. Suppose that f : (1, 1)  S1 ! M is a smooth map with the property that f (0, ) ¼ g and f ( , 0) ¼ g(0). Each f (s, ) is a map from S1 to M, so the assignment s ! f (s, ) can be viewed as a 1-parameter family of deformations of the based loop g. Granted this view, then each

13.8

Holonomy and curvature

s 2 (1, 1) version of f (s, ) has its corresponding holonomy; and these together give a smooth, G-equivariant map hA; f ð Þ : ð1; 1Þ  Pjgð0Þ ! G: This map is G-equivariant in the sense that hA, f ( )(pg1) ¼ g hA,f ( )(p) g1. For s 2 (1, 1) near 0, one can write hA; f ðsÞ ðpÞ ¼ hA;g ðpÞ emðs; pÞ where m(s, p) 2 lie(G) is such that m(s, pg1) ¼ g m(s, p) g1. Moreover, m is smooth on some neighborhood of 0  Pjg(0). Note that this can be done because of what is said in Chapter 5.5 about the exponential map giving a diffeomorphism from a neighborhood of the origin in lie(G) to a neighborhood of the identity element in G. Thus, if h 2 G is any given element, then the map n ! h en defines a diffeomorphism from a neighorhood of the origin in lie(G) onto a neighborhood of h in G. Of interest is the derivative with respect to the variable s 2 (1, 1) at s ¼ 0 of the map m( , ). The fourth property of h is an expression for this derivative in terms of the curvature of the connection A. This expression for this derivative requires some additional notation. For the latter, introduce fs ¼ ðf* ]s] Þjs¼0 and ft ¼ ðf* ]t] Þjs¼0 ¼ g

] * ]t

to denote the respective sections of g*TM ! S1 that are obtained using the

>

>

differential of f . Meanwhile, introduce g*P ! S1 to denote the pull-back bundle. The pull-back of FA via g, this g*FA, is a section over S1 of the vector bundle N 2 g*T*M) over S1; thus g*FA(f s, f t) is a section over S1 of (g*P ad lie(G)) the bundle g*P ad lie(G). (Here, 2 T*M should be thought of as the subbundle N of antisymmetric homomorphisms in the bundle Hom(T*M T*M; R).) As a section of g*P ad lie(G), the element g*FA(f s, f t) can be viewed as an equivariant map from g*P to lie(G). Meanwhile, the lift gA,( ) defines an isomorphism g*Pj[0,2p) ! [0, 2p)  Pg(0), and this isomorphism identifies g*FA(f s, f t) as a G-equivariant map from [0, 2p]  Pjg(0) to lie(G). I use g*A FA(f s, f t) to denote the latter map. This map is G-equivariant in the sense that ðgA*FA ð f s ; f t ÞÞjðt; pg1 Þ ¼ gðgA*FA ð f s ; f t ÞÞjðt; pÞ g1 : With the digression now over, what follows is the fourth property of the holonomy: Property 13.6 The derivative at s ¼ 0 of m is 

2p  ð ]  m  ¼  ðg*ðF A A ðfs ; ft ÞÞjðt;pÞ dt: s¼0 ]s 0

Note for the future that this formula tells us that m is independent of s if A is a flat connection. This implies, in particular, that it has the same value for homotopic loops.

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The derivation of this formula is a nice exercise unwinding all of the definitions. As such, it is also left to the reader save for the following remarks. Remark 13.7 Let U  M denote an open set with an isomorphism j: PjU ! U  G. Write j*A as g1dg þ g1a g. As in N Chapter 12.7, write 2 j*FA as the G-equivariant section over U  G of lie(G) T*MjU whose 1 value at (x, g) is (x, g (FA)U g) where (FA)U is the lie(G)-valued 2-form given by (da þ a ∧ a).

>

162

Remark 13.8 Suppose that g(t0) 2 U. It follows from what is said in Chapter 11.6 that the horizontal lift of f (s, ) for t near t0 and s near 0 is mapped by j to the path in the product bundle U  G that send t to (f (s, t)), g(s, t)) where g(s, ) obeys the equation g1

] g ¼ að f t Þjf ðs; Þ: ]t

Here, a( f t) is obtained by evaluating the lie(G)-valued 1-form a on the vector f t ¼ f ]s] : *

Remark 13.9 Take the covariant derivative of this last equation with respect to the variable s, and use the formula in Chapter 12.6 that expresses FA in terms of commutators of covariant derivatives to j*FA to obtain an equation for the covariant derivative in the t-direction of ðg1 ]s] gÞ. Integrate this last equation to get the desired equation for ð]s] mÞjs¼0.

13.9

Proof of the classification theorem for flat connections

The proof is given below in four parts.

13.9.1

Part 1

The correspondence is given by a map of the following sort: Fix an orientation and base point for S1. Also, fix a base point x 2 M. Let P ! M denote a principal G-bundle, and let A denote a flat connection on P. Fix a point p 2 Pjx. As explained momentarily, the assignment of hA,g to a loop g 2 Lx defines a homomorphism, rA, from p1(M) to G. The image of rA in Hom(p1(M); G)/G gives the desired correspondence. To start the explanation, remark that it is a consequence of the four properties of the holonomy listed in the preceding section of this chapter that hA,g defines a homomorphism from p1(M) to G. The image in Hom(p1(M); G)/G of

13.9

Proof of the classification theorem for flat connections

this homomorphism does not depend on the chosen point p 2 Pjx ; this is a consequence of the fact that hA,g(pg1) ¼ g1hA,g(p)g. To go further, suppose that j: P0 ! P is a bundle isomorphism. With the point p 2 Pjx chosen, set p0 ¼ j1(p). It is then the case that hj*A,g(p0 ) ¼ hA,g(p). This is because j1(gA,p) has horizontal tangent vectors, and so must equal gj*A,p0 . As a consequence, the image of hA,g in Hom(p1(M); G)/G depends only on the equivalence class of the pair (P, A) in FM,G.

13.9.2

Part 2

Suppose next that pairs (P, A) and (P0, A0 ) define equivalence classes in FM,G with the same image in Hom(p1(M); G))/G. What follows is a construction of an isomorphism j: P0 ! P with the property that j*A ¼ A0 . To start, fix points p 2 Pjx and p0 2 P0 jx. Let y 2 M, and choose a path g: [0, 1] ! M such that g(0) ¼ x and g(1) ¼ y. Define jg : Pjy ! P0 jy as follows: Recall that the lift gA,( ) defines an isomorphism from Pjx to Pjy and that gA0 ,( ) defines one from P0 jx to P0 jy. This understood, a G-equivariant map from Pjy to G is defined by writing 1 gA,( ) (py) ¼ p h(py). The map py ! h(py) is G-equivariant in the sense that h(pyg1) ¼ g1h(py)g. This understood, set jg ðpy Þ ¼ gA0 ; p0 ð1Þhðpy Þ: If g0 is homotopic to g, then jg0 ¼ jg as can be seen using Property 13.6. To see about other choices for g0, suppose that n is a loop based at x, and let g0 ¼ g n denote the concatenation of n and g. All homotopy classes of paths from x to y can be realized by some loop of this sort. Given that (P, A) and (P0, A0 ) have the same image in Hom(p1(M); G) /G, it follows that jg0 ¼ jg for such g0 . This understood, set j: Pjy ! P0 jy to equal jg for any choice of g. To see that j is smooth, consider a point y0 very near y, and use a path that is obtained by concatenating first g and then a short arc (say a geodesic with respect to some fixed metric) from y to y0. You can use local trivializations of P and P0 near y and the differential equation from Chapter 11.6 for the horizontal lift to see that j is smooth. The fact that j*A ¼ A0 follows from the fact that the construction implies that a horizontal lift to P as defined by A of any given path g is mapped by j to a horizontal lift of g to P0 as defined by A0 . This implies that the differential of j maps HA  TP to HA0  TP0. This is to say that A0 ¼ j*(A).

13.9.3

Part 3

It remains now to prove that all elements in Hom(p1(M); G)/G can be obtained from elements in FM,G. To do this, reintroduce the universal covering space for

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˜ . Suppose that r is a given homomorphism from p1(M) to G. M, denoted by M ˜ r G ¼ (M ˜  G)/ where the Use r to construct the principal G-bundle Pr ¼ M equivalence relation identifies pair (y˜, g) with (cz(y˜), r(z) g) for z 2 p1(M). This bundle Pr comes with a canonical flat connection, Ar, which is defined as ˜ G follows: The lie(G)-valued 1-form A0 ¼ g1dg on the product bundle M is pulled back to itself by any diffeomorphism that sends (y˜, g) to (cz(y˜), r(z) g) for z 2 p1(M). This understood, this A0 is the pull-back of a lie(G)-valued 1-form on TPr. The latter is the connection Ar. This connection is flat because ˜ of its curvature is that of A0 and A0 is flat. the pull-back to M

13.9.4

Part 4

To see what the holonomy is for the connection Ar around loops in M based at x, let g: [0, 1] ! M denote a loop based at x and let z denote its image in p1(M). ˜ that projects to x. Then g has a unique lift as a path Fix a point x˜ 2 M ˜ y˜: [0, 1] ! M that maps 0 to x˜ and maps 1 to c1 ˜ ) where z1 is the inverse z (x of z in p1(M). The parallel transport of Ar along g starting from the image in Pr of a point (x˜, g) is, by definition, the parallel transport of A0 along the path y˜. This gives the point (c1 ˜ ), g). The equivalence class of this point in Pr is not z (x that of (x˜, g), but rather that of (x˜, g (g1r(z)g)). This implies that the image of the representation r in Hom(p1(M); G)/G is the same as that defined by the pair (Pr, Ar) via the map from FM,G to Hom(p1(M); G)/G.

Appendix 13.1

Smoothing maps

Suppose that M and N are smooth manifolds, and that f: M ! N is a continuous map. As it turns out, the map f can be approximated by a smooth map. The discussion that follows considers only the case when M is compact. As a warm-up, consider first the case where N ¼ Rn. Fix a finite, open cover of M by coordinate charts. Let U denote this cover; when U 2 U, let jU : U ! Rm denote the corresponding coordinate chart map. Here, m denotes the dimension of M. Fix also a subordinate partition of unity for this cover. These are described in Appendix 1.2. Denote the partition of unity by {wU}U2U. Finally, fix a small positive number r. This number r will determine the accuracy of the approximation. One more item is needed, Ðthis the function w: [0, 1) ! [0, 1] that is described in Appendix 1.2. Let c ¼ Rm wðjxjÞdm x. Granted all of the preceding, set fr : M ! Rn equal to the function

Appendix 13.1

p ! f r ðpÞ ¼

X U2U

wU ðpÞrn c1

ð Rm

Smoothing maps

m fðj1 U ðxÞÞwðjx  jU ðpÞj=rÞd x:

You can think of the value of fr at any given p 2 M as an average of the values of f over the part of M with distance on the order of r or less from p as defined by a given Riemannian metric. In any event, fr is smooth because the various U 2 U versions of jU and wU are smooth. To see how close fr comes to f, note that a change of integration variable allows fr(p)  f(p) to be written as ð X m n 1 w ðpÞr c ðfðj1 f r ðpÞ  fðpÞ ¼ U ðx þ jU ðpÞÞÞ  fðpÞÞwðjxj=rÞd x: U2U U Rm

Since f is continuous and M is compact, the following can be said: Given e > 0, there exists re > 0 such that wU(p) jf(jU1(x þ jU(p)))  f(p)j  e wU(p) when jxj < re . This being the case, if r < re , then the equation above for fr(p)  f(p) implies that jfr  fj < e at all points in M if r < re . The construction just given has the property that fr will in general differ from f at all points in M. However it may be the case that f fails to be smooth only on a subset of M; and in this case, it can be useful to have fr differ from f only near this subspace. A simple modification of the construction will have this property. Let Z  M denote a compact subspace. Suppose the goal is to modify f only on some given open neighborhood of Z. To do so, let UZ  M denote the given 0 neighborhood and let UZ  UZ denote an open set with compact closure in UZ that also contains Z. Choose the open cover U so that it has the form 0 U ¼ {U0} ¨ UZ , where U0 ¼ MUZ , and where UZ consists of open sets with compact closure in UZ. It is not necessary that U0 come with a coordinate chart map. Such a map won’t exist in general. However, each U 2 UZ should come with such a map. Now define fr by the rule: ð X m n 1 wU ðpÞr c fðj1 p ! f r ðpÞ ¼ wU0 fðpÞ þ U ðxÞÞwðjx  jU ðpÞj=rÞd x: U2U z

Rm

This smoothing agrees with f on MUZ. Turn now to the case where N is a smooth manifold. Let n denote the dimension of N. Because M is compact, so f(M)  N is a compact set. This understood, the constructions that are described in Chapter 1.3 can be used to obtain the following: An integer d n, an open set Y  N that contains f(M), and an embedding c: Y ! Rd whose image is a submanifold of Rd. Granted this data, then c(f) defines a map from M to Rd. I can smooth this map in the manner just described so as to get a smooth map, c(f)r , from M to Rd that comes as close as desired to c(f). However, in general, c(f)r will not map into

165

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c(Y), but just near to c(Y) in Rd when r is small. So, more must be done to obtain an approximation to f as a map into Y. To start this task, introduce the (dn)-dimensional vector bundle p: E ! Y whose fiber at any given point y is the subspace of vectors v 2 Rd that are orthogonal to all vectors tangent at c(p) to c(Y). This is to say that E  Y  Rd consists of the pairs (y, v) such that v is orthogonal to the image of TYjy via Chapter 5.3’s push-forward map, c*. Note that E can also be viewed as the pull-back via c of the normal bundle to the submanifold c(Y). (The normal bundle to a submanifold is defined in Chapter 4.2.) Define next a map, C: E ! Rd, by the rule C(p, v) ¼ c(p) þ v. By construction, the differential of this map along the zero section is an isomorphism from TEj0 to TRdjc(Y). This fact can be used with what is said in Chapter 1.4 about immersions and submersions to find r0 > 0 and an open neighborhood Y0  Y of f(M) with the following properties: Let E0  E denote the open subset consisting of the points (y, v) with y 2 Y0 and jvj < r0. Then C restricts to E0 as a diffeomorphism of E0 onto an open neighborhood of c(Y0). With the preceding understood, remark next that if r is sufficiently small, then the smoothing c(f)r of c(f) will map M into C(E0). Granted such a choice for r, then the assignment p ! C1(c(f)r(p)) defines a smooth map from M to E. This the case, then a smoothing of f is given by the map p ! fr(p) ¼ p(C1(c(f)r(p))).

Appendix 13.2

The proof of the Frobenius theorem

This appendix gives a proof of the Frobenius theorem (Theorem 13.1). The proof is by induction on the dimension n. To set up the induction, remark first that the Frobenius theorem is automatically true for n ¼ 1. This said, suppose that n > 1 and that the Frobenius theorem is true for all dimensions from 1 through n1. The five steps that follow explain how to prove the Frobenius theorem for dimension n given this assumption.

Step 1 Fix a point p 2 X. Since H is a vector bundle in its own right, there is a neighborhood, U, of p with a vector bundle isomorphism j: HjU ! U  Rm. Introduce {e1, . . . , en} to denote a basis for Rm. The vectors fields fj1 e1 ; j1 e2 ; : : : ; j1 em g

Appendix 13.2

The proof of the Frobenius theorem

span H at each point of U. By taking a somewhat smaller neighborhood of p if necessary, I can require that U be a coordinate chart for X, and so U comes with a diffeomorphism c: U ! Rn that sends p to the origin. Compose this diffeomorphism with the action on Rn of a matrix in Gl(n; R) if necessary so that (v1 ¼ c*j1e1, v2 ¼ c*j1e2, . . . , vm ¼ c*j1em) at the origin 0 2 Rn are the first m elements of the coordinate basis, thus ð]x] 1 ; : : :; ]x]m Þ. This is assumed in what follows. As v1 j0 ¼ ]x] 1 , there is a neighborhood of 0 on which the vector

fields ðv1 ; ]x]2 ; : : : ;

] ]xn Þ

are linearly independent.

Step 2 This step constructs a diffeomorphism from a neighborhood of the origin in Rn to Rn that fixes 0 and whose differential maps the first coordinate vector, ]x] i , to v1. The desired diffeomorphism is denoted by s. The map s is defined by the following rule:  s (0, x2, . . . , xn) ¼ (0, x2, . . . , xn)  s (x1, x2, . . . , xn) is the point at time x1 on the integral curve of v1 that starts when x1 ¼ 0 at the point (0, x2, . . . , xn). Invoke the vector field theorem (Theorem 8.2) to construct the map s. To see that s is a diffeomorphism near 0, use the fact that v1 j0 ¼ ]x] 1 to conclude that the differential of s at the origin is the identity. Granted that such is the case, invoke the inverse function theorem (Theorem 1.1) to conclude that s defines a diffeomorphism on some neighborhood of the origin. With s as just described, the inverse map is a diffeomorphism whose differential sends v1 to ]x]1 and maps 0 to 0 with differential equal the identity at 0. Compose the diffeomorphism from Step 1 with this inverse and so obtain a new diffeomorphism of a neighborhood of p in X with a neighborhood of 0 in Rn that takes p to 0, whose differential on a neighborhood of p sends j1e1 to ]x] 1 , and whose differential at p sends (j1e2, . . . , j1em) to ð]x] 2 ; : : : ; ]x]m Þ. Use c now to denote this new coordinate chart map; and for each k 2 {2, . . . , m}, use vk now to denote the vector field given by the push-forward via this new version of c of j1ek.

Step 3 Since H is involutive, h iany k 2 {2, . . . , m} version ofPthe vector field defined by the commutator ]x] 1 ; vk can be written as ck ]x] 1 þ 2jm ckj vj where ck and the various j 2 {2, . . . , m} versions of cjk are functions defined on a neighborhood of 0 in Rn. I use c in what follows to denote the map from this neighborhood

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of the origin to the vector space M(m  1, R) of (m  1)  (m  1) matrices whose (k, j) component is ckþ1,jþ1. Use the vector field theorem (Theorem 8.2) to construct a map, g, from a somewhat smaller neighborhood of 0 2 Rn to Gl(m  1; R) that obeys ] g ¼ gc and gjx1 ¼0 ¼ i: ]x1 !

Let c denote the map from a neighborhood of 0 to Rm1 whose k’th component is ckþ1. Use g and this map to define a second such map, b, by the rule xð1

!

!

b ðx1 ; x2 ; : : :; xn Þ ¼ ðgc Þðs;x2 ; : : :; xn Þds: 0

Step 4 For k 2 {2, . . . , m}, use wk to denote the vector field X ] wk ¼ g v  bk1 : 2jm k1;j1 j ]x1 Since bk1 ¼ 0 where x1 ¼ 0 and g ¼ i where x1 ¼ 0, the set f]x] 1 ; w2 ; : : :; wm g of vector fields span c*H at each point on some neighborhood of 0 in Rn. This new spanning set for c*H has been introduced for the following reason: For each k 2 {2, . . . , m}, the following is true: h i  x]1 ; wk ¼ 0;  wk is orthogonal to ]x] 1 ,  wk j0 ¼ ]x] k : Indeed, the maps g and b are defined precisely so as to guarantee this.

Step 5 The first two bullets above are fancy ways of saying the following: When written in terms of the coordinate basis f]x] i g1jn , the vector field wk has the form wk ¼ Ak2

] ] þ : : : þ Akn ]x2 ]xn

where each coefficient function Aki is a function only of the coordinates (x2, . . . , xn). As a consequence, the x1 coordinate is superfluous. In particular, the collection {w2, . . . , wm} can be viewed as a set of linearly independent

Additional reading

vector fields defined on a neighborhood of 0 2 Rn1. Moreover, they span an involutive subbundle, H0  TRn1, of dimension m  1. This understood, I now invoke the induction hypothesis to complete the proof of the theorem.

Additional reading  Geometry, Topology and Physics, Mikio Nakahara, Taylor and Francis, 2003.  Topology and Geometry, Glen E. Bredo, Springer, 2009.  Differential Manifolds, Lawrence Conlon, Birkha¨user, 2008.  Geometry of Differential Forms, Shigeyuki Morita, American Mathematical Society, 2001.  Geometry of Four-Manifolds, S. K. Donaldson and P. B. Kronheimer, Oxford University Press, 1997.  Algebraic Topology: A First Course, M. J. Greenberg and J. R. Harper, Benjamin/Cummings, 1981.

169

14

Curvature polynomials and characteristic classes

This chapter explains how the curvatures of connections can be used to construct De Rham cohomology classes that distinguish isomorphism classes of vector bundles and principal bundles. These classes are known as characteristic classes. They are of great import to much of modern topology, differential topology, differential geometry, algebraic geometry and theoretical physics.

14.1

The Bianchi Identity

The definition of the characteristic classes involves new machinery, a part of which is introduced here. To start, let P ! M denote a principal G-bundle and let A denote a connection on P. The curvature 2-form of A is again denoted by N 2 FA, this, as I hope you recall, is a section of (P ad lie(G)) ∧ T*M. As explained below, this 2-form always obeys the following differential equation: dA FA ¼ 0; N

N

where dA: C1(M; (P ad lie(G)) ∧2 T*M) ! C1(M; (P ad lie(G)) ∧3 T*M) is the exterior covariant derivative (from Chapter 12.4) that is defined by the connection A. This identity is known as the Bianchi identity. There are various proofs of this identity; these depending on your point of view about connections. Here is a quick proof: Let r denote a representation of G into some n  1 version of either Gl(n; R) or Gl(n; C). Use V in what follows to denote either Rn or Cn as the case may be and use Gl(V) to denote the corresponding space of invertible, real or complex n  n matrices. Recall from Chapter 12.4 that the connection A induces an exterior covariant derivative, dA, for the sections of N (P r V) ∧p T*M, and that FA enters in the following way: Let s denote a section of P r V. Then dA2s is equal to r*(FA) s where r*: lie(G) ! lie(Gl(V)) is

14.2

Characteristic forms

the differential at the identity of r. This understood, one can write dA3s in two ways. Here is the first: dA3s ¼ dA ðdA2sÞ ¼ r* ðdA FA Þs þ r* ðFA Þ ∧ dA s: On the other hand, dA3s ¼ dA2ðdA sÞ ¼ r* ðFA Þ ∧ dA s: These two equations are compatible if and only if r*(dAFA) ¼ 0. Take V such that r* is injective to see that dAFA ¼ 0. What follows is a second proof of the Bianchi identity. Recall the definition of FA that uses local isomorphisms of P with a product bundle. By way of a reminder, let U  M denote an open set with an isomorphism j: PjU ! U  G. Write the pull-back (j1)*A at (x, g) as g1dg þ g1aUg where aU is a lie(G)-valued 1-form on U. Meanwhile, j supplies an isomorphism (also denoted by j) from (P ad lie(G))jU to U  lie(G), and with respect to this isomorphism, the curvature FA appears as the section x ! (x, (FA)U) of the N bundle (U  lie(G)) T*MjU where ðFA ÞU ¼ daU þ aU ∧ aU : Keeping this in mind, recall now the exterior covariant derivative for sections of the bundle P ad lie(G): If s is a section of this bundle, then j(s) can be written as the section of U  lie(G) that sends x to the pair (x, sU(x)) where sU N maps U to lie(G). Then j(dAs) is the section of (U  lie(G)) T*M that sends x ! (x, (dAs)U) with ðdA sÞU ¼ dsU þ aU sU  sU aU ¼ dsU þ ½aU; sU where [,] indicates the commutator of matrices. Here, sUaU and aUsU are matrices of 1-forms, these obtained by multiplying the matrix sU and matrix of 1-forms aU in the indicated order. Granted this, it then follows that j(dAFA) N is the section x ! (x, (dAFA)U) of (U  lie(G)) ∧3 T*M where ðdA FA ÞU ¼ dðdaU þ aU ∧ aU Þ þ aU ∧ ðdaU þ aU ∧ aU Þ  ðdaU þ aU ∧ aU Þ ∧ aU : This is zero. (Keep in mind that d(aU ∧ aU) ¼ daU ∧ aU  aU ∧ daU.)

14.2

Characteristic forms

Suppose that m > 0 and that r: G ! Gl(n; C) is a representation of G on V ¼ Cm. This induces the representation ad(r) of G on the vector space M(n; C) whereby a

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given g 2 G acts on a matrix m 2 M(n; C) to give the matrix r(g)mr(g)1. Let tr: M(n; C) ! C denote the trace. This induces a bundle homomorphism trr: PadðrÞ Mðn; CÞ ! M  C: To see that this is well defined, note that a given point m 2 (P ad(r) M(n; C)) is an equivalence class of pairs (p, m) with p 2 P and m 2 M(n; C) where the equivalence relation identifies such a pair with (pg1, r(g)m r(g)1) when g 2 G. Given that the trace obeys tr(gmg1) ¼ tr(m) for any pair m 2 M(n; C) and g 2 Gl(n; C), it follows that trr gives the same value on any two representatives of a given equivalence class. The homomorphism trr as just defined extends to give trr :



Pad lie ðGÞ

O

∧p T*MÞ ! ∧p T*MC N

for any given integer p. Here, T*MC ¼ T*M C. This extension is defined by first specifying how it acts on the decomposable elements: This action sends N m o with m 2 P ad(r) M(n; C) and o 2 ∧p T*M having the same base point as m to trr(m) o. The action of trr on a sum of decomposable elements is uniquely determined by the linearity requirement. Granted all of this, suppose now that A is a connection on P. Fix an integer p. N The p-fold wedge product of r*(FA) is a section of (P ad(r) M(n; C)) ∧2p T*M. This understood, define the form sp ðr; FA Þ ¼

 i p 2p

trr ðr* ðFA Þ ∧ : : : ∧ r* ðFA ÞÞ:

The following two comments elaborate on the meaning of the expression on the right hand side of this last equation. N

Comment 14.1 Any given element z 2 (P ad(r) M(n; C)) ∧2p T*M can be N written as a finite sum z ¼ k mk ok where each mk 2 P ad(r) M(n; C), where each ok 2∧2p T*M, and where all sit over the same base point in M. Then trr(z) ¼ k trr(mk) ok. Comment 14.2 With regards to the meaning of r*(FA) ∧ . . . ∧ r*(FA), suppose that a and b are respectively matrix valued q and r forms on M. This is to N q ∧ T*M and b is a section say that a is a section of (P ad(r) M(n; C)) N of (P ad(r) M(n; C)) ∧r T*M. Then their product a ∧ b is the section N of (P ad(r) M(n; C)) ∧qþr T*M that is defined as follows: Write a as a N sum of decomposable elements, thus as a ¼ k ak mk; and likewise write N b ¼ i bi ni. Here, each ak and bi is a section of P ad(r) M(n; C) and each mk and ni is a q-form and r-form, respectively. Then

14.2

a ∧ b ¼ Sk;i ðak bi Þ

O

Characteristic forms

ðmk ∧ ni Þ:

This last formula is used in an iterative fashion to define r*(FA) ∧ . . . ∧ r*(FA) by taking a above to be r*(FA) and b to be the p  1 fold version of r*(FA) ∧ . . . ∧ r*(FA). As defined, sp(r, FA) is a C-valued 2p-form on M. However, if r maps G into U(n), then sp(r, FA) is in fact an R-valued 2p-form. This is so by virtue of the fact that (r*(FA)){ ¼ r*(FA). Indeed, it follows from this last identity that the complex conjugate of the right-hand side of the trr(r*(FA) ∧ . . . ∧ r*(FA)) differs from the original by p factors of 1, these coming from the replacement of (r*(FA)){ with r*(FA). Keep in mind here that the trace on Mðn; CÞ obeys trðmÞ ¼ tr m{ Þ. Meanwhile, there are another p factors of 1 from the complex conjugation of ð2pi Þp. The form sp has the following two important features:  The form sp is a closed 2p-form. This is to say that dsp(r, FA) ¼ 0.  The cohomology class of sp(r, FA) does not depend on A: If A and A0 are two connections on P, then sp(r, FA0 )  sp(r, FA) ¼ da where a is a (2p1)-form. To see about the top bullet, note first that d trr ðmÞ ¼ trr ðdA mÞ for any section m of P ad(r) M(n; C) and any connection A on P. You can check that this is true on an open set U  M with an isomorphism to U  G. Note in addition that dA ðm ∧ m0 Þ ¼ dA m ∧ m0 þ ð1Þq m ∧ dA m0 N

when A is any connection on P, m is any section of (P ad(r) M(n; C)) ∧q T*M, N 0 and m0 one of (P ad(r) M(n; C)) ∧q T*M. Again, this can be verified by using the formula for dA on an open set U as above. Granted these points, it follows directly that dtrr ðFA ∧ ::: ∧FA Þ¼ trr ðdA FA ∧FA ∧ ::: ∧FA Þþtrr ðFA ∧dA FA ∧ ::: ∧FA Þþ ::: ¼ 0 as each term in this sum has a factor of dAFA which is zero because of the Bianchi identity. To see about the lower bullet, introduce [0, 1]  M and the map p: [0, 1]  M ! M given by the projection. In the meantime, recall that A0 can be written as A0 ¼ A þ p*a where p: P ! M is the projection and a is N T*M. Granted this formula for A0 , then the a section of (P ad lie(G)) ˆ ¼ A þ ta for t 2 (0, 1) defines a connection on the pull-back bundle formula A p*P ! (0, 1)  M. The curvature 2-form of Aˆ at any given t 2 [0, 1] is

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FA^ jt ¼ dt ∧ a þ FA t where FAt is the curvature on M of the connection A þ ta. Note in particular that the latter has no dt component. The 2p-form sp(r, FAˆ) is closed on (0, 1)  M. Write this form in the manner of Chapter 12.2 as sp ðr; FA^ Þ ¼ dt ∧ ap1 þ ap where ap1 is a t-dependent (2p  1)-form on M and ap is a t-dependent 2p-form on M. The latter is sp(r, FAt). The fact that sr is closed requires that @ ap ¼ dap1 : @t Integrate this with respect to t 2 [0, 1] to see that 01 1 ð sp ðr; FA0 Þ  sp ðr; FA Þ ¼ d@ ap1 jt dtA: 0

This last equation tells us that sp(r, FA0 )  sp(r, FA) can be written as da.

14.3

Characteristic classes: Part 1 2p

Since the form sp(r, FA) is closed, it defines a class in H De Rham(M). This class is independent of the chosen connection, and so depends only on the chosen principal bundle and the representation r. The sp version is denoted here by sp(Pr). As explained next, if P and P0 are isomorphic principal bundles, then the corresponding sp cohomology classes are identical. Indeed, this is a consequence of the following observations. Suppose that j: P ! P0 is a principal G-bundle isomorphism and that A is a connection on P0. Then j*A is a connection on P. This understood, it is enough to prove that sp ðr; FA Þ ¼ sp ðr; Fj*A Þ: This last identity can be seen by viewing P and P0 over an open set U  M where there are isomorphisms jU : PjU ! U  G and also jU0 : P0 jU ! U  G. The composition jU0  j  jU appears as the map from U  G to itself that sends any given pair (x, g) to the pair (x, hU(x)g) where hU maps U into G. The image via jU of the curvature of A over U is defined by the lie(G)-valued 2-form (FA)U, and the image via jU0 of the curvature of j*A is defined by that of the lie(G)-valued 2-form (Fj*A)U. In particular, these two lie(G)-valued 2-forms are determined one from the other from the formula

14.4

Characteristic classes: Part 2

ðFj*A ÞU ¼ hU ðFA ÞU hU 1 : Given that the trace obeys tr(hmh1) ¼ tr(m), the preceding relation between curvatures implies that the two sp forms are identical. Two conclusions follow immediately:  Principal G-bundles P and P0 are not isomorphic if there exists a representa2p tion r of G and an even integer p such that sp(Pr) 6¼ sp(Pr0 ) in H De Rham(M).  A principal G-bundle does not have a flat connection if there exists a representation r of G and an even integer p such that sp(Pr) 6¼ 0. Any given version of sp is an example of a characteristic class for the bundle P.

14.4

Characteristic classes: Part 2

The construction described in the previous two sections of this chapter can be generalized along the following lines: Fix n  1 and let f : M(n; C) ! C denote any smooth function with the property that f (gmg1) ¼ f (m). Such a function is said to be ad-invariant. Examples are tr(mp) for p a positive integer. Suppose that f is an ad-invariant function. Fix t  0 and write the Taylor’s expansion of f (tm) as f 0 þ f1 ðmÞt þ f2 ðmÞt2 þ f3 ðmÞt3 þ : : : : Each f k is also an ad-invariant function on M(n; C), but one that is a homogeneous polynomial of degree p in the entries of m. This is to say that f k is the composition f k ¼ fk   k

Nk where the notation is as follows: First, k : M(n; C) ! M(n; C) sends N N m to its k-fold tensor product m . . . m. Meanwhile, fk is a linear map N N N fk : k M(n; C) ! C that is ad-invariant. This is to say that fk(m1 . . . mk) N N ¼ fk(gm1g1 . . . gmkg1) for any g 2 G. An example is the map that sends N N (m1 . . . mk) to tr(m1 . . . mk). The function f k is completely determined by fk. Here are two important examples of ad-invariant functions:  cðtmÞ ¼ detði þ 2pi tmÞ     chðtmÞ ¼ tr exp 2pi t m . What follows is a basic result concerning the vector space of ad-invariant functions.

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Theorem 14.3 (the ad-invariant function theorem) The vector space of real analytic, ad-invariant functions that are homogeneous of a given positive degree p is the C-linear span of the set {tr(mk1) . . . tr(mk4) : k1þ . . . þ kq ¼ p}. Appendix 14.1 contains a proof of this assertion. By way of an example, c(t m) and ch(t m) given above can be written as  cðmÞ ¼ 1 þ 2pi t trðmÞ  8p12 ðtrðmÞ2  trðm2 ÞÞt2 þ : : : :  chðmÞ ¼ 1 þ Sk1 k!1 ð2pi Þk trðmk Þtk . With the preceding as background, let P ! M denote a principal G-bundle and let A denote a given connection on P. Let r: G ! Gl(n; C) denote a homomorphism of G. Suppose in what follows that f is an ad-invariant function Lon M(n; C). Then f and r can be used to construct the section f (r, A) of 1kdim(M) ∧2k T*M, this being a differential form that can be thought of as f (r*(FA)). To give a precise definition, write f ðtmÞ ¼ f0 þ t f 1 ðmÞ þ 12 t2 f2 ðmÞ þ : : : where each f k(·) is a ad-invariant, homogeneous polynomial of degree k in the entries of the matrix. Granted this expansion, define f ðr* ðFA ÞÞ ¼ 1 þ f 1 ðr* ðFA ÞÞ þ f 2 ðr* ðFA ÞÞ þ : : : where f k(r*(FA)) is the 2k-form on M that is defined as follows: View N N N N Nk 2 r*FA . . . r*FA as a section of (P ad(r) ( k M(n; C))) ( (∧ T *M). N Then antisymmetrize the 2-form part to obtain a section of (P ad(r) ( k M(n; C))) N 2k ∧ T*M. To complete the definition, recall that f k is determined by a function N fk from k M(n; C) to C by the rule f k ¼ fk  k. Take fk and apply it along each fiber to obtain a C-valued section of ∧2k T*M. If allowed to modify previously introduced notation, I denote the result as fk(r*FA ∧ . . . ∧ r*FA). By way of an example to make sure things are understood, suppose that r*(FA) is written on some open set in M as m1o1 þ m2o2 where o1 and o2 are 2-forms, and m1 and m2 are sections of P ad(r) M(n; C). Then  f 1(r*(FA)) ¼ f1(m1) o1 þ f1(m2) o2.  f 2(r*(FA)) ¼ f2(m1, m1) o1 ∧ o1 þ (f2(m1, m2) þ f2(m2, m1)) o1 ∧ o2 þ f2(m2, m2) o2 ∧ o2. Note in particular that only the symmetric part of f2 appears. This is a general phenomenon: Only the completely symmetric part of any given fk appears in the formula for f k(r*(FA)). I leave it to you to convince yourself that this is the case. As a consequence, it is sufficient to consider only symmetric, ad-invariant functions when defining characteristic classes. The arguments given in the previous section of this chapter concerning the case when f (m) ¼ tr(mk) can be repeated with only cosmetic modifications to prove the following:

14.5

Characteristic classes for complex vector bundles and the Chern classes

 Each f k(r*(FA)) is a closed form.  The cohomology class of f k(r*(FA)) does not depend on the chosen connection A. This class thus depends only on P and r. It is denoted in what follows by f k(Pr).  If P and P0 are isomorphic principal bundles, then the cohomology classes f k(Pr) and f k(Pr0 ) are the same. Each f k(Pr) is also an example of a characteristic class. The last three sections of this chapter give examples of bundles with nonzero characteristic classes. It is worth keeping in mind that characteristic classes behave naturally with respect to pull-back by smooth maps between manifolds. To elaborate, suppose that X is a given smooth manifold and p: P ! X is a principal G-bundle. Suppose that n  1 and that r: G ! Gl(n; C) is a representation. Use r to define a characteristic class f k(Pr). Now let M denote a second manifold and c: M ! X a smooth map. The characteristic class f k((c*P)r) 2 H*De Rham(M) is the pull-back in De Rham cohomology via c of the corresponding class fk(Pr) 2 H*De Rham(X). This is to say that the cohomology class of f k((c*P)r) is represented by the pull-back via c of a closed 2p-form on X that represents the cohomology class of f k(Pr). Here is why: Fix a connection A on P. Then c*A is a connection on c*P and its curvature 2-form is given by Fc*A ¼ c*FA. As a parenthetical remark, note that any given map c: M ! X induces by p p pull-back a linear map c*: HDe Rham(X) ! HDe Rham(M). This is defined as p follows: Let z denote a given class in HDe Rham(X). Choose a closed p-form, o, p to represent z. Then c*z 2 HDe Rham(M) is represented by c*o. This pull-back of De Rham cohomology is well defined because the pull-back of a closed form is closed and the pull-back of an exact form is exact. This is to say that dc* ¼ c*d.

14.5

Characteristic classes for complex vector bundles and the Chern classes

Let p: E ! M denote a complex vector bundle; thus the fiber of E is Cn for some positive integer n. It is customary to associate characteristic classes directly to E rather than going through the construction of a principal bundle that has E as an associated bundle. (For example, one could use the principal bundle PGl(E) of frames in E.) This is done as follows: Fix a covariant derivative  for sections of E, and then introduce from Chapter 12.4 the exterior covariant derivate d and define the curvature F from d2. Let f denote an ad-invariant function on M(n; C). The covariant derivative  and an integer k determine the

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corresponding closed 2k-form f k(F). Here again, this 2k form is obtained Nk N N Nk from the section F of ( k End(E)) ( (∧2 T*M)) by applying N the antisymmetrization homomorphism to obtain a section of ( k End(E)) N 2k ∧ T*M and then applying the linear map fk defined by f to obtain the desired C-valued 2k-form. The arguments given in the preceding sections of this chapter can be repeated with only notational changes to see that f k(F) is a closed 2k-form on M; and that it’s cohomology class does not depend on the covariant derivative. The latter is denoted by f k(E). Note that if E and E0 are isomorphic vector bundles, then f k(E) is the same as f k(E0 ). The cohomology class f k(E) is an example of a characteristic class for the vector bundle E. The principal bundle and vector bundle notions of these closed forms are essentially the same. To elaborate, suppose that P is a principal G-bundle, that n is a positive integer, and that r: G ! Gl(n; C) is representation. Let E denote the associated bundle P r Cn. Then f k(Pr) and f k(E) are identical. Characteristic classes for vector bundles pull-back via smooth maps just as those for principal bundles, and for essentially the same reason. To elaborate, suppose that X is a smooth manifold, and that p: E ! X a complex vector bundle of dimension n. Let M denote another smooth manifold and let c: M ! X denote a smooth map. Chapter 6.11 explains how to define the pull-back bundle c*E ! M. Then any given characteristic class of c*E is the pull-back via c*: H*De Rham(X) ! H*De Rham(M) of the corresponding characteristic class for E. This last observation has the following consequence: Let M denote a given manifold and E a given complex vector bundle of dimension n. As noted in Chapter 6.11, if m is a sufficiently large integer, then there is a map, c, from M to the complex Grassmannian GrC(m; n) and a vector bundle isomorphism between E and the pull-back via c of the tautological, rank n vector bundle over GrC(m; n). Thus, all of E’s characteristic classes are pulled back by c . This can be interpreted as follows: Characteristic classes are pull-backs of “universal characteristic classes” that live in the cohomology of all sufficiently large m versions of GrC(m; n). In the case when f ðmÞ ¼ cðmÞ ¼ detði þ 2pi mÞ, the corresponding characteristic class f k is denoted by ck(E) and it is called the k’th Chern class of E. The Chern classes are useful by virtue of the fact that ck(E) ¼ 0 if the fiber dimension of E is less than k. This is so because function m ! detði þ 2pi mÞ is a polynomial in the coefficients of m of degree equal to the fiber dimension. For example, if E has fiber dimension 1, then only c1(E) can be nonzero. If E has fiber dimension 2, then only c1(E) and c2(E) can be nonzero. To say more about the utility of the Chern classes, I first remind you that the N wedge product of forms endows H*De Rham(M) ¼ 1kdim(M) HkDe Rham(M) with the structure of an algebra. This algebra is defined as follows: Let a1 and

14.6

Characteristic classes for real vector bundles and the Pontryagin classes p

a2 denote two cohomology classes, say with a1 2 HDeRham(M) and with q a2 2 HDe Rham(M). To see about their product, let o1 and o2 denote any two closed forms that represent these classes. The class a1 a2 is represented by the form o1 ∧ o2. To see that this form is closed, note that dðo1 ∧ o2 Þ ¼ do1 ∧ o2 þ ð1Þp o1 ∧ do2 : To see that the class of o1 ∧ o2 does not depend on the chosen representative, consider that if o10 ¼ o1 þ db, then o01 ∧ o2 ¼ o1 ∧ o2 þ dðb ∧ o2 Þ: The algebra so defined on H*De Rham is such that a1a2 ¼ (1)pq a2a1. What was said above about Chern classes can be restated as follows: The set of all characteristic cohomology classes is generated as a subalgebra in H*De Rham(M) by the Chern classes. This is to say that any characteristic class that is defined by a real analytic, ad-invariant function on M(n; C) is a sum of products in H*De Rham of these Chern classes. The appendix to the chapter explains why this is so. The classes that arise using chðmÞ ¼ trðexpð 2pi mÞÞ result in what are called Chern character classes.

14.6

Characteristic classes for real vector bundles and the Pontryagin classes

Characteristic classes for bundles E ! M with fiber Rn can also be defined without directly referencing a principal bundle. Here is how this is done: First construct the complexification of E as done in Chapter 6.3. By way of a reminder, this is the rank n complex bundle, EC , whose underlying real bundle N is (M  R2) E. This bundle has an almost complex structure, j, that comes from the almost complex structure   0 1 on R2 : j0 ¼ 1 0 N

To elaborate, j is first defined on decomposable vectors so as to send a pair z N e with z 2 R2 and e 2 E to j0z e, and it is then extended to the whole of N (M  R2) E by linearity. With R2 viewed as C, then j0 acts as multiplication by i; and the C-action on EC is such that s 2 C sends the decomposable element N N z e to s z e. A covariant derivative, , for sections of E induces a covariant derivative N E that commutes with the action of j. The for sections of (M  R2) N induced covariant derivative acts so as to send a decomposable section, z s

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N

to z s þ dz s. Because  commutes with j, it defines a covariant on EC. This induced covariant derivative is denoted by C . The characteristic classes of EC as defined by any ad-invariant function on End(Cn) are viewed as characteristic classes of the original Rn-bundle E. An important example of this construction gives the Pontryagin classes. The k’th Pontryagin class of E is the 4k-dimensional chomology class, pk ðEÞ ¼ ð1Þk c2k ðEC Þ: Note in this regard that the Chern classes ck(EC) vanish if k is odd; thus there is no motivation to use the odd k Chern classes of EC to define characteristic classes of E. Here is why these odd k classes vanish: Fix a fiber matrix on E. This done, one can find a covariant derivative on E that comes from a connection on the principal bundle PO(E) of orthonormal frames in E. Let f now denote an adinvariant function on End(Cn). I claim that the odd k versions of f k(F) of the corresponding covariant derivative on EC are all zero. This is because m can be written as gmg1 for some g 2 SO(n). To see why this is, remark that m has some 2j purely imaginary eigenvalues, these in complex conjugate pairs. Taking the real eigenvectors and the real and complex parts of the complex eigenvectors finds an orthonormal basis for Rn of the form fu1 ; : : : ; un2j ; v1 ; : : : ; v2j g with the property that muq ¼ 0 for q 2 {1, . . . , n2j}, and mv2q1 ¼ aq u2q and mv2q ¼ aq u2q1 for q 2 {1, . . . , j} with aq > 0. This understood, let g denote the matrix in SO(n) that switches v2q1 with v2q for each q 2 {1, . . . , j} and either fixes the span of {uq}q=1, . . . ,n2j if j is even, or fixes {uq}2qn2j and sends uj to uj if j is odd. This matrix g has the desired properties.

14.7

Examples of bundles with nonzero Chern classes

What follows are some examples where the Chern classes are nonzero. Example 14.4 This is a principal U(1)-bundle over S2. View S2 as the unit sphere in R3. Reintroduce the Pauli matrices t1, t2, and t3, from Chapter 6.4. Use them as in Chapter 6.4 to define the complex, rank 1 bundle E1  S2  C2 to be the set of pairs ðx ¼ðx1 ; x2 ; x3 Þ; vÞ that obey xj tj v ¼ iv: The bundle E1 has a Hermitian metric, this coming from the Hermitian metric on S2. This is to say that the norm of (x, v) 2 E1 is jvj. Let UE ! S2 denote the bundle of unitary frames in E1. As noted in Chapter 10.3, the space UE when viewed only as a smooth manifold is just S3.

14.7

Examples of bundles with nonzero Chern classes

The projection P: S2  C2 ! E at x 2 S2 is 12 ð1  ixj tj Þ. A covariant derivative on E1 is given as follows: View a section, s, of E1 as a map, also denoted by s, from S2 to C2 with the property that s(x) 2 E1jx at each x 2 S2  C2. Granted this view, set s ¼ Pds. As explained in Chapter 12.5, the curvature of this covariant derivative is F ¼ PdP ∧ dPP: To be more explicit, note that dP ¼  2i dxj tj . Given that the Pauli matrices multiply as the quaternions (t1t1 ¼ 1, t1t2 ¼ t3, etc), it follows that dP ∧ dP ¼

1 i dx ∧ dxj eijk tk ; 4

where eijk is completely antisymmetric under interchange of any of its indices; and it obeys e123 ¼ 1. To see what this is at a given x 2 S2, fix an oriented, orthonormal frame (x, u, v) for R3, thus (u, v) are an orthonormal frame for TS2jx. This done, then the matrix-valued 2-form dP ∧ dP evaluates on (v, u) to give 12 eijk vi uj tk ¼ 12 xk tk . Since xktk acts as multiplication by i on E1, it follows that F(u, v) is just multiplication by 2i. Thus, the Chern form c1(E1; ) 1 . Note that this is independent evaluates on the frame, (v, u) to give 2pi ð2i Þ ¼  4p 2 of x 2 S . As a consequence, the integral of c1(E1, ) over S2 is 1 as S2 has area 4p. This tells us that the first Chern class of E is nonzero in H2De Rham(S2). Indeed, were c1(E1) ¼ 0, then c1(E1; ) could be written as da for some 1-form a. Keep in mind here that there is a well-defined notion of the integral of an n-form on an n-dimensional, compact and oriented manifold, and this integral is zero if and only if the n-form is the exterior derivative of an (n  1)-form. Appendix 14.2 discusses this subject of integration on manifolds for those needing a refresher course. The preceding calculation tells us that E1 is not isomorphic to S2  C. The calculation also tells us that the principal U(1)-bundle UE has no flat connections. Example 14.5 The tangent bundle to S2 can be viewed as the subbundle in S2  R3 that consists of the pairs (x, v) such that x·v ¼ 0. Here, ‘·’ denotes the dot product. The orthogonal projection P: S2  R3 ! TS2 sends a pair (x, v) to (x, v  x (x·v)). As explained in Chapter 11.3, this gives a covariant derivative, . Recall that  is defined by viewing a section S2  R3, and so a section of TS2 as a map from S2 to R3 that lands in TS2 at each point. This understood, , sends such a map, s, to s ¼ Pds ¼ ds þ xdxTs. The curvature of this covariant derivative is F ¼ dx ∧ dxT:

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To see what this is, fix x 2 S2, and then fix a pair (u, v) of orthonormal vectors in R3 such that (x, u, v) is an oriented, orthonormal basis for R3. Then F(u, v) is the endomorphism F ðu; vÞij ¼ ui vj  vi uj : Meanwhile, TS2 has an almost complex structure, j, that acts on any given point (x, v) to give (x, x  v), where ‘’ here denotes the cross product on R3. Let T1,0 ! S2 denote the corresponding complex, rank 1 vector bundle. Let x 2 S2 and let (u, v) be as above. Then T1,0jx is spanned by z ¼ u  iv. A calculation will verify that  commutes with j and so defines a covariant derivative, denoted here by  also, on the space of sections of T1,0. The curvature of this connection sends z to (F)ijzj ¼ izi. Thus, F acts as multiplication by i. 1 This implies that c1(T1,0; F) is 2p times the standard area 2-form on S2, and so 2 its integral over S is equal to 2. This implies that T1,0 is neither the trivial C-bundle nor isomorphic to E1. Example 14.6 What follows is an application of pull-back to the computation of Chern classes. Let   R3 denote a compact, embedded surface. Use x ! n(x) 2 R3 to denote the unit length, outward pointing normal vector to  at x. The tangent space to  can be viewed as the subbundle of   R3 consisting of the pairs (x, u) such that n(x)·u ¼ 0. As noted in Chapter 6.5, this bundle has the almost complex structure j that acts to send a point (x, u) to the point (x, n(x)  u). Use T1,0 to denote the resulting rank 1 complex vector bundle. This bundle is, in fact, the pull-back from S2 of the bundle T1,0 from the previous example. Indeed, it is precisely n*T1,0 where n:  ! S2 is the map x ! n (x). This said, it follows that c1(T1,0) ¼ n*c1(T1,0). The following proposition is needed to exploit this last observation. Proposition 14.7 Suppose that X is a compact, oriented, n-dimensional manifold and suppose that M is another such manifold.  Suppose that c: M ! X is a smooth map. There exists Ðan integer pÐ with the following property: Let o denote a n-form on X. Then M c*o ¼ p X o . This integer p is said here to be the degree of c.  Homotopic maps from M to X have the same degree. The proof of this proposition is given in Appendix 14.3. To use this proposition, orient  so that (n(x), u, v) is an oriented frame for TR3 at a point x if (u, v) is an oriented frame for T at x. This is the case, for example if u is any given nonzero vector and v ¼ n  u. Use this orientation with the preceding proposition to define the notion of the degree of a map from  to S2. Appendix 14.3 describes an embedding of  into R3 whose version of the map x ! n(x) has degree equal to 1 genus(). Granted that such is

14.7

Examples of bundles with nonzero Chern classes

the case, it follows from the proposition that any 2-form on  that represents the corresponding version of c1(T1,0) integrates to zero when  is the torus, and it integrates to 2  2g when  is a surface of genus equal to g; this being a surface as depicted in Chapter 1.3 with g holes. As I now argue, this last conclusion holds for any version of T1,0 whose defining almost complex structure, j, has the following property: If p 2  is any given point and u 2 Tjp is any given nonzero vector, then (u, j·u) is an oriented frame for T. Let j denote an almost complex structure on T of the sort just described and let T1,0 denote the corresponding complex 1-dimensional vector bundle. I use the next lemma to prove this claim that the integral over  of any representative of c1(T1,0) is equal to 2  2g. Lemma 14.8 Let M denote a compact, manifold and p: ER ! M a real vector bundle with even dimensional fiber. Suppose that t ! jt is a smoothly varying, 1-parameter family of almost complex structures on ER, this parametrized by t 2 [0, 1]. For each such t, let Et denote the complex vector bundle that is defined from ER via defined by jt. Then each t 2 [0, 1] version of Et is isomorphic to E0 as a complex vector bundle. In particular, the Chern classes do not change as t varies. Given the example in Appendix 14.3, the proof of the claim about the integral of c1(T1,0S) follows from the preceding lemma with a proof of the following assertion: Any two complex structures on T that define the same orientation for T are connected by a path of complex structures. This fact is proved momentarily. What follows is the proof of the lemma. Proof of Lemma 14.8 Change the parametrization if necessary so that the map t ! jt is constant near t ¼ 0 and also near t ¼ 1. Let p: (0, 1)  M ! M denote the projection. The smooth family t ! jt defines an almost complex structure on p*ER such that the resulting complex vector bundle, E, restricts to any given t 2 (0, 1) version of {t}  M as the bundle Et. Fix a connection, A on E. Parallel transport by A along the fibers of p defines a complex linear isomorphism between E(0) and any t 2 (0, 1) version of E(t). Now consider the assertion that any two almost complex structures on T that give the same orientation are connected by a path of almost complex structures. To start, suppose that j is a given almost complex structure on T. Introduce T1,0j to denote j’s version of T1,0. This is, by definition, the subbundle of the N complexification T C on which j acts as multiplication by i. Let j0 denote another almost complex structure. Let p denote a point in , and e 2 T1,0jjp. N Then j0 ·e is a vector in T C, and as such, it can be written as j0 e ¼ a e þ b e¯ with a and b complex numbers. Act by j0 on this expression for j0 ·e to see that

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Þb e : e ¼ða2  jbj2 Þ e þ ða þ a Indeed, this follows because j0 2 ¼ 1 and because j0 e¯ is the complex conjugate of j0 e. This last equation requires that a ¼ i(1 þ jbj2)1/2. Moreover, the þ sign must be taken in order that (v, j0 v) define the correct orientation. Each point in  has a corresponding b, and the assignment of b to the point in  defines a section over  of Hom(T1,0j; T0,1j). The converse of this observation is also true: Any given section, b, of the bundle Hom(T1,0j; T0,1j) defines an almost complex structure, jb, where jb acts on e 2 T1,0j by the rule jbe ¼ i(1 þ jbj2)1/2 e þ b e¯. To see jb as an endomorphism of T, write e as u1  iu2 with u1 and u2 vectors in T. Likewise, write b as b1 þ ib2 with b1 and b2 real. Then jb u1 ¼ðð1 þ jbj2 Þ1=2  b2 Þu2 þ b1 u1 and jb u2 ¼ ðð1 þ jbj2 Þ1=2 þ b2 Þu1  b1 u2 : With the preceding understood, write j0 ¼ jb as above. Then the 1-parameter family {t ! j(1t)b}t2[0.1] exhibits a path of almost complex structures that starts when t ¼ 0 at j and ends when t ¼ 1 at j0 . Example 14.9 Let  denote a compact, oriented surface. This means that ∧2 TM has a distinguished section up to multiplication by a positive real number. See Chapter 10.3. Construct a complex, rank 1 vector bundle E !  as follows: Choose k  1 and a set of k distinct points  ¼ {p1, . . . , pk}  . Assign to each point p 2  an integer mp. Fix disjoint coordinate patches around the points in . Given p 2 , let Up   denote the corresponding coordinate patch, and let cp : Up ! R2 a coordinate chart sending p to 0. Assume that the differential of cp is orientation preserving in the sense that it sends the distinguished section of ∧2 TM to a positive multiple of @x@ 1 ∧ @x@ 2 . I use (r, y) for standard polar coordinates on R2. Thus x1 ¼ rcosy and x2 ¼ rsiny. The bundle E is defined over the cover of M given by the set U0 ¼  and {Up}p2. Over each such set, E is defined to be the product bundle. The only intersections are between U0 and Up for each p 2 . The cp image of U0 Up is the subset R2{0} of R2. This understood, define the bundle E so that when x 2 U0 Up, then the point (x, v) 2 Up  C corresponds to the point (x, g0,p(x)v) 2 U0  C where g0,p(x) is such that if x ¼ cp(r, y) then g0,p ¼ eimpy. I claim that the integral over  of c1(E) is equal to –p2 mp. To verify this claim, I will fix a convenient covariant derivative for E whose corresponding first Chern form is easy to integrate. To start, fix a smooth function, w, from [0, 1) to [0, 1] which is equal to 0 near 0 and equal to 1 on [1, 1). I now view w as the function on R2 that sends (r, y) to w(r). ¨

¨

184

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The covariant derivative, , on E is defined as follows: Identify E’s restriction to Up with Up  C so as to view a section of E on Up as a map x ! sp(x) from Up to C. Use cp to view s on Up as a C-valued 1-form, (s)p, on R2. Then  is defined so that ðs Þp ¼ dsp þ i mp sp w dy: Identify E’s restriction to U0 as U0  C so as to view s over U0 as a map, x ! s0(x) from U0 to C. By way of a reminder, s0 over U0 Up is given in terms of sp by the formula ¨

s0 ðxÞ ¼ eimp y sp ðxÞ where y is the angular coordinate of cp(x). View s over U0 as a C-valued 1-form, this denoted by (s)0. Then  is defined so that ðs Þ0 ¼ ds0  ∑p2∧ i mp s0 c*p ðð1  wÞdyÞ: You can check that (s)0 ¼ eimpy(s)p at any given point x 2 U0 Up. In particular,  as defined is a bonafide covariant derivative. The preceding formulae for  show that F ¼ d2 ¼ 0 except in the sets Up. In any such set, the formula above for (sp) implies that (cp1)*F is the Gl(1; C)valued 2-form on R2 given in polar coordinates by ¨

ðF Þp ¼ imp w0 dr ∧ dy: d w. It follows that c1(E; ) has support only in the sets Up, and in Here, w0 ¼ dr 1 any such set, it pulls back via cp1 to the 2-form  2p mp w0 dr dy. As a consequence, its integral over Up is equal to mp. Thus, its integral over  is P  p2 mp.

Example 14.10 Fix n  1 and recall that CPn is the Grassmannian of 1-dimensional complex vector spaces in Cnþ1 as defined in Chapter 6.8. An alternate view of CPn as S2nþ1/S1 described in Example 10.3 and Example 10.8. By way of a reminder, view S2nþ1 as the unit sphere in Cnþ1 and view S1 as the unit sphere in C. Have S1 act on Cnþ1 so that any given l 2 S1 sends any given (nþ 1)-tuple (z1, . . . , znþ1) to (lz1, . . . , lznþ1). This action preserves S2nþ1. The action is free, and the quotient is the space CPn. As explained in Chapters 10.5 and 10.6, the projection p: S2nþ1 ! CPn defines a principal S1 ¼ U(1) bundle. As the Lie algebra of U(1) is the imaginary complex numbers, a connection on this bundle is an i-valued 1-form on S2nþ1 that is pulled back to itself by the action of any l 2 S1 and pairs with the vector field

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  @ @ v ¼ i ∑1k1 zk  zk @zk @ zk to give i. Note that the vector field v spans the kernel of p* and is such that if f is any given function on S2nþ1 and z =(z1, . . . , znþ1) 2 S2nþ1, then d it it dt f ðe z1 ; : : : ; e znþ1 Þ at t ¼ 0 is equal to v(f )jz. Granted what was just said, the i-valued 1-form A¼

1 ∑ ðz dzk  zk dzk Þ 2 1knþ1 k

from Cnþ1 is a connection when viewed as a 1-form on S2nþ1. The pull-back via the projection p: S2nþ1 ! CPn of the curvature 2-form FA is the restriction to S2nþ1 of the form p*FA ¼ ∑1knþ1 d zk ∧ dzk on Cnþ1. Note that this form on S2nþ1 annihilates the vector v since p*FA(v, ·) is P P the 1-form d( 1knþ1 jzkj2) and 1knþ1 jzkj2 ¼ 1 on S2nþ1. The resulting 2-form that defines the first Chern class of this line bundle pulls back to S2nþ1 as P zk ∧ dzk . the restriction from Cnþ1 of the form o ¼ 2pi 1knþ1 d  To see that the first Chern class is nonzero, it is enough to note that the i (2nþ1)-form 2pi A ∧ ok ¼ 2p A ∧ o ∧ : : : ∧ o is a constant multiple of the 2nþ1 volume form on S . Thus, its integral is nonzero. This can happen if and n only if the k’th power of the Chern class c1(E) is nonzero in H2k De Rham(CP ). k 2k n Thus, c1(E) 6¼ 0. Indeed, suppose that c1(E) ¼ 0 in H De Rham(CP ). Then the form ð2pi Þk FA ∧ : : : ∧FA can be written as da with a a (2k 1)-form on CPn. Were such the case, then ok ¼ d(p*a). The integral of 2pi A ∧ ok would then equal that of 2pi A ∧ dðp*aÞ which is   i i A ∧ p*a þ dA ∧ p*a: d 2p 2p However, dA ¼ p*FA and so the integral of 2pi A ∧ ok over S2nþ1 would equal the integral of p*o ∧ p*a. However the latter is p*(o ∧ a), which is to say that it is the pull-back of a degree (2n þ 1)-form on CPn. All such forms are zero since dim(CPn) ¼ 2n. Example 14.11 This example considers principal SU(2)-bundles over an oriented 4-dimensional Riemannian manifold. Let M denote the manifold. The bundle will be specified by an open cover with suitable transition functions on the overlapping regions between sets of the cover. To start, fix a nonempty, finite set   M. For each p 2 , fix an open set Up that contains p and that has an orientation-preserving diffeomorphism cp : Up ! R4 sending p to 0. I use y ¼ (y1, y2, y3, y4) for the Euclidean coordinates of R4. To say that M is oriented

14.7

Examples of bundles with nonzero Chern classes

is to say that the bundle ∧4 T*M has a given nowhere zero section (see Chapter 10.3). To say that cp is orientation preserving is to say that the pull-back via cp of the 4-form dy1 ∧ dy2 ∧ dy3 ∧ dy4 from R4 is a positive multiple of this section. Choose these charts so that the Up is disjoint from Up0 when p 6¼ p0 . Let U0 ¼ M. The only transition functions to consider are those for U0 Up for p 2 . Such a function is given by a map, g0,p : R4{0} ! SU(2), by the rule g0,p(x) ¼ gp(cp(x)). To specify gp , I first choose an integer, mp. Set g: R4{0} ! SU(2) to be the map that sends a point y 2 R4 to the matrix   1 y4 þ iy3 iy1  y2 jyj : iy1 þy2 y4 iy3 ¨

Now set gp ¼ gm where gm ¼ g . . . g if m > 0, or g1 . . . g1 if m < 0, or the identity matrix i if m ¼ 0. Let r: SU(2) ! Gl(2; C) denote the defining representation. I claim that the Chern class c2(Pr) is nonzero, and that it evaluates on M’s fundamental class as p2 mp. To do this, I need to choose a convenient connection. For this purpose, I take the function w from Example 14.9. The connection A is defined by giving, for each p 2 , a lie(SU(2))-valued 1-form ap on Up, and also a 1 1 lie(SU(2))-valued 1-form a0 on U0. These must be such that ap ¼ g0p dg0p þ g0p 1 a0 g0p on U0 Up. This understood, I take (cp )*ap to be the lie(SU(2))-valued 1-form on R4{0} given by ¨

mp dðgmp Þ: ðc1 p Þ*ap ¼ wðjyjÞ g

Meanwhile, I take a0 ¼ 0 except on Up, and here  m mp g p: ðc1 p Þ*a0 ¼ wðjyj  1Þdðg Since the trace of a matrix in lie(SU(2)) is zero, the 2-form that defines the second Chern class is 1 trðFA ∧ FA Þ: 8p2 I argue momentarily that the second Chern class is nonzero if p2 mp is nonzero. This follows from the assertion that the integral of the preceding form over M is p2 mp. This shows, in particular, that this construction of principal bundles gives nonisomorphic bundles when the respective sums differ. To see what the integral of this Chern form is, note first that the form in question is zero except in Up for p 2 . Moreover, the support of this form is contained in the inverse image via cp of the unit radius ball about the origin in R4. This understood, it is enough to compute, for each p, the following integral

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1 8p2

ð trðFap ∧ Fap Þ jyj1

where Fap ¼ dap þ ap ∧ ap. The computation that follows finds this integral equal to mp. To see about this integral, I will employ the following fact: Let G denote a Lie group and a denote a lie(G)-valued 1-form defined on some open set in any given manifold. Introduce Fa ¼ da þ a ∧ a. Then    2 trðFa ∧ Fa Þ ¼ d tr a ∧ da þ a ∧ a ∧ a : 3 This identity will be proved in Chapter 14.9. Take the a ¼ ap version of this identity and integrate by parts to see that the integral at the end of the previous paragraph is equal to  ð  1 2 tr a ∧ da þ ∧ a ∧ a a p p p p p : 8p2 3 jyj¼1

A computation finds the integrand equal 1  trðgmp dðgmp Þ ∧ gmp dðgmp Þ ∧ gmp dðgmp ÞÞ: 3 This in turn is the pull-back via the map from S3 to SU(2) given by y ! gmp jy of the form 1  trðg1 dg ∧ g1 dg ∧ g1 dgÞ 3 on SU(2). This latest 3-form is invariant under both left and right multiplication. This said, its evaluation at the identity i 2 SU(2) finds it equal to 4 times the volume form on SU(2) that comes by identifying SU(2) with the unit radius sphere in R4. Granted all of this, it then follows that the integral of 8p12 trðFA ∧ FA Þ over Up is given by the product of three factors. The first is 8p1 2 . The second is 4 times the volume of the 3-sphere, which is 8p2. The third is the degree of the map y ! gmpjy from S3 to SU(2), or what is the same thing, the integral over S3 of the pull-back via this map of the 2p1 2 times the volume form. The first two factors give an overall factor of 1, and so the contribution from Up to the integral of the form that gives the second Chern class is just 1 times the pull-back of 2p12 times the pull-back volume form by the map y ! gmp jy. As explained momentarily, the latter number is mp. Granted this, then the integral of the second Chern class does have the predicted value p2 mp.

14.8

14.8

The degree of the map g ! gm from SU(2) to itself

The degree of the map g ! gm from SU(2) to itself

The map y ! gmjy from S3 to SU(2) where  1 y4 þ iy3 gðyÞ ¼ jyj iy1 þy2

iy1  y2 y4 iy3



becomes the map g ! gm from SU(2) to SU(2) when S3 is identified in the usual way with SU(2). This map has degree m. More to the point, what is proved next is that the integral of the pull-back of the volume form on SU(2), O¼

1 trðg1 dg ∧ g1 dg ∧ g1 dgÞ; 12

by this map is m times the volume of S3. To see this, start with the case m ¼ 0. The map in this case is g ! i, the constant map to the identity. Thus, the map pulls back the volume form as zero. In the case m ¼ 1, the map is g ! g which is the identity map, so the pull-back of the volume form is itself. To see about m ¼ 1, note that the pull-back of g1dg by the map g ! g1 is dgg1. As a consequence, this map pulls back O as ð1Þ3

1 trðdgg1 ∧ dgg1 ∧ dgg1 Þ 12

which is seen to be O using the cyclic property of the trace. Note in this regard that if a is a matrix-valued p-form and b is a matrix-valued q-form, then trðabÞ ¼ ð1Þpq trðbaÞ: It follows from this and the fact that gm ¼ (gm)1 that it is sufficient for the purposes at hand to show that the map g ! gm for m positive pulls O back to a form whose integral over S3 is m times that of O. To do so, suppose that this has been established for all integers m 2 {1, . . . , n}. To consider the case m ¼ n þ 1, write the map g ! gn as gh. Then the map g ! gh pulls back g1dg as h1g1dgh þ h1dh. This is a matrix-valued 1-form. Multiplying matrices and using the aforementioned cyclic property of the trace finds that the map g ! gnþ1 pulls back O as 1 1 trðg1 dg ∧ g1 dg ∧ g1 dgÞ þ trðh1 dh ∧ h1 dh ∧ h1 dhÞ 12 12 1 1 þ trðg1 dg ∧ g1 dg ∧ dhh1 Þ þ trðdhh1 ∧ dhh1 ∧ g1 dgÞ: 4 4 Now the two left-most terms above are O and h*O, respectively, so the sum of their integrals over SU(2) is n þ 1 times the volume of SU(2). As for the two

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right-most terms, note that d(g1) ¼ g1dgg1 and d(h1) ¼ h1dhh1. Therefore, the sum of the two right-most terms above can be written as   1 1 1 d  trðg dg ∧ dhh Þ : 4 Thus, the integral of the two right-most terms over SU(2) is zero.

14.9

A Chern–Simons form

The identity asserted in Chapter 14.8 can be proved by directly computing both sides. What follows is another approach that can be used for other purposes. Let t denote the coordinate on [0, 1]. Then Aˆ ¼ g1dg þ tg1ag is a connection on the principal bundle ([0, 1]  U)  G. Its curvature 2-form is FAˆ ¼ tda þ dt ∧ a þ t2a ∧ a. This understood, the 4-form on [0, 1]  U given by tr(FAˆ ∧ FAˆ) is closed. This form can be written in the manner introduced in Chapter 12.2 as dt ∧ a3 þ a4 where a3 ¼ 2trðta ∧ da þ t2 a ∧ a ∧ aÞ; and where a4 has no factor of dt. At any given t 2 [0, 1], the 4-form a4 is tr(FAˆ(t) ∧ FAˆ(t)) where Aˆ(t) is the connection on the principal U  G on U given by Aˆ(t) ¼ g1dg þ t g1ag. For example, Aˆ(0) is the product, flat connection and Aˆ(1) ¼ g1dg þ g1ag. In any event, since tr(FAˆ ∧ FAˆ) is a closed form, it follows @ that @t a4 ¼ da3 . Integrate this identity with respect to t 2 [0, 1] to see that    2 trðFAð1Þ ∧FAð1Þ Þ ¼ trðFAð0Þ ∧FAð0Þ Þ þ d tr a ∧ da þ a ∧ a ∧ a : ^ ^ ^ ^ 3 This last equality gives the identity in Chapter 14.8 by virtue of the fact that FAˆ(0) ¼ 0 and FAˆ(1) ¼ da þ a ∧ a. The 3-form   1 2  2 tr ^ a ∧d ^ a þ ^a ∧ ^a ∧ ^a 8p 3 is an example of Chern–Simons form.

Appendix 14.1

The ad-invariant functions on M(n; C)

What follows first is a proof of the ad-invariant function theorem (Theorem 14.3) from Chapter 14.4. Recall that this theorem asserts that the real analytic, ad-invariant functions on M(n; C) are generated as an algebra by the functions

Appendix 14.1

The ad-invariant functions on M(n; C)

from the set {m ! tr(mk)}k1. A few words are then said about the assertion made at the end of Chapter 14.5 to the effect that the algebra of characteristic classes of a given vector bundle are generated by its Chern classes. Proof of the ad-invariant function theorem Note to start that any ad-invariant function on the vector space M(n; C) must be a symmetric function of the eigenvalues of the matrix. The reason being that an open dense set of matrices are diagonalizable over C. (To be diagonalizable means that the matrix is conjugate to a diagonal matrix.) You can find a proof of this fact in almost any undergraduate book on linear algebra. As there are n eigenvalues, and granted the observation of the preceding q paragraph, it is enough to know that the set {1kn lj} q=0,1, . . . n of functions n of the coordinates (l1, . . . , ln) for C generates the algebra of symmetric, analytic functions on Cn. To see that such is the case, view the coordinates Q l ¼ (l1, . . . , ln) as the roots of the polynomial z ! hl(z) ¼ k (z  lk). The coefficients of this n’th order polynomial are symmetric functions of the roots. This is to say that when hl(z) is written out as hl ðzÞ ¼ zn  a1 zn1 þ : : : þ ð1Þn an ; then each aj is a symmetric function of the roots. For example, a1 ¼ k lk and Q an ¼ k lk. If two polynomials have the same coefficients, then they have the same roots, so this implies that the functions {a1, . . . , an} of the roots generate the algebra of symmetric functions of the coordinates on Cn. Granted Q this, consider now the function u ! g(u) on C given by g(u) ¼ j (1 þ lju) ¼ anun þ . . . þ a1u þ 1. As is clear, the coefficients of g(u) are just this set of symmetric functions {a1, . . . , an} of the roots. It follows as a consequence that the coefficients of the power series expansion of   ln gðuÞ ¼ ∑ j lnð1 þ lj uÞ provide a generating set for the algebra of symmetric functions of the coordinates of Cn. This is because the k’th order term in the expansion of g(u) consists of sums of product of the set {aj : j < k} plus a nonzero multiple of ak. This said, note that the power series for ln(g(u)) is   m ln gðuÞ ¼ ∑m  1 ð1Þmþ1 m1 ð∑1jn lm j Þ u : The preceding formula shows that the coefficients are proportional to the symmetric functions {jlm j }m  0. This is what was claimed by the theorem. The final remarks in this appendix concern the assertion from Chapter 14.5 that the algebra of characteristic classes of a given vector bundle are generated by its Chern classes. Given what was said above in the proof of the

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ad-invariant function theorem, this assertion is an immediate consequence of the following two observations: First, the coefficients of the polynomial u ! cðuÞ ¼ Pj ð1  2pi lj uÞ generate the algebra of symmetric functions on Cn. Second, the coefficients of this polynomial are the very same functions (viewed as ad-invariant functions on M(n; C)) that define the Chern classes.

Appendix 14.2

Integration on manifolds

This appendix reviews the basics of integration on compact manifolds. The following proposition summarizes the story. Proposition 14.12 Let M denote a compact, oriented, n-dimensional manifold. There is a well-defined notion of the integral of an n-form on M. An n-form that is not identically zero and defines the given orientation where it is nonzero has positive integral. On the other hand, the integral of the exterior derivative of an (n1)-form is zero. An example involving spheres is given after the proof of this proposition. What follows is an immediate corollary. Corollary 14.13 Let M denote a compact, oriented, n-dimensional manifold. Then the De Rham cohomology in dimension n has dimension at least 1. Indeed, any n-form whose integral is nonzero projects with nonzero image to HnDe Rham(M). Proof of Corollary 14.13 As there are no nonzero (nþ1)-forms, the De Rham cohomology in dimension n is the vector space C1(M; ∧n T*M)/im(d). This understood, the corollary follows from the proposition’s assertion that the integration of the exterior derivative of any given (n1)-form is zero. As it turns out, an n-form can be written as the exterior derivative of an (n1)-form if and only if its integral is zero. This implies that HnDe Rham(M) has dimension 1. Proof of Proposition 14.12 The proof has four parts. The first part reviews integration over domains in Rn. Part 2 states three important properties of the integral. Part 3 uses what is said in Part 1 to give a definition of the integral of an n-form that employs a cover of M by coordinate charts. Part 4 proves that the value of the integral as defined in Part 3 does not depend on the chosen coordinate chart cover. Part 5 proves that a nontrivial n-form that defines the orientation where it is nonzero has positive integral, and that the integral of the exterior derivative of an (n1)-form is zero.

Appendix 14.2

Integration on manifolds

Part 1: This part gives a definition of the integral of a compactly supported function in Rn. The integral here is the n-dimensional version of the 1-variable calculus definition via the “midpoint approximation” that you might remember from your second semester of calculus. To start, suppose that f is a smooth function on Rn which is zero outside of some bounded set. What follows defines the integral of f. Because the set where f is nonzero is bounded, there exists R > 0 such that this set is contained in the n-dimensional cube where the coordinates (x1, . . . , xn) satisfy R  xi  R for each i 2 {1, . . . , n}. Fix N  1 and introduce N to denote the set of vectors of the form k ¼ (k1, . . . , kn) where each i 2 {1, . . . , n} version of ki is drawn from the set {1, . . . , N}. This set N has Nn elements. For each k 2 N , introduce fk to denote the value of f at the point whose coordinate entries obeys   2ki  1  1 for each i 2 f1; : : :; ng: xi ¼ R N As explained next, the following limit exists: limN!1 ð2RÞn Nn ∑k2N f k :

ð*Þ

This limit is defined to be the integral of the function f. I denote the latter by ð fdn x: Rn

To see that this limit exists, consider the difference between the N and N þ L version for any given integer L > 1. For k 2 k, let &R,k  Rn denote the n-dimensional cube where the coordinates are such that each entry obeys R

2ki  2  N 2ki  N  xi  R : N N

The center of &R,k is the point that defines fk. The union of these cubes is n [R, R), this the cube that contains the support of f. Meanwhile, pair-wise distinct cubes are disjoint. The replacement of N by N þ L when evaluating ð2RÞn Nn ∑k2N f k involves the evaluation of f at (2L)n points in any given version of &R,k. Given that f is differentiable, the value of f at any such point differs from fk by no more than a cf RN1, where cf is the supremum as x ranges in n [R, R] of jdfj. This being the case, it follows that the N þ L version of the sum written above differs from the N version by at most cf RN1. This last fact proves that the sequence of such sums, as indexed by N, is a Cauchy sequence and so has a unique limit. As noted, this limit is defined to be the integral of f.

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Part 2: Three properties of the integral of f are used in what follows. Ð  Positivity: Rn fdn x > 0 if f is nonnegative and strictly greater than zero at some point. Ð @vi  The integral of a divergence is zero: Rn fdn x ¼0 if f can be written as 1in @x i where (v1, . . . , vn) are compactly supported functions on Rn.  Change of variable formula: Suppose that V  Rn is an open set and let c: V ! Rn denote an embedding whose image contains the set where f is nonzero. Then ð ð ðc*fÞjdetðc* Þjdn x ¼ f dn x: Rn

Rn

The claim made by the first bullet is, I trust, self-evident. To see about the second bullet, it is enough to prove that the integral of each term in the sum is @v1 zero. I consider the case @x . The arguments for the other terms differ only in 1 notation. To start the argument, fix N large. Write a given vector k 2 N as a pair (k1, k0 ) with k1 2 {1, . . . , n} and k0 2 n1 {1, . . . , N}. Define x ¼ (xi, x2, . . . , xn) to be the center point of the n-dimensional cube &R,k, thus the point in Rn whose i’th coordinate is given by xi ¼ R(2ki  1)/N. Write x ¼ (x1, x0 ) with x0 ¼ (x1, . . . , xn). Now recall the definition of the partial derivative to write          @v1 N 2R 2R 0 0 ðx1 ; x Þ ¼ þe v1 k1 ; x  v 1 k 1  1 ; x0 @x1 2R N N where jej is bounded by an N-independent multiple of N1 . With the preceding understood, note next that the sum (*) can be ordered so that one fixes k0 and then sums over all values of k1 before going to another value of k0 . Keeping this in mind, it follows that the sum of the various k1 versions of the expression @v1 cancel but for the error term e. For any given fixed k0 , there are at above for @x 1 most N such error terms, so their sum is bounded by some k0 and N independent number. As there are Nn1 distinct vectors k0 to consider, it follows that the totality of these error terms contribute to (*) at most some N-independent multiple of RnNn times this factor Nn1; thus at most an N-independent multiple of RnN1. The N ! 1 limit of these error terms is therefore zero. The change of variable can be seen by using (*) with the following observation: Suppose that x 2 Rn is in the domain of c. Fix r > 0 and let & denote a cube of side length r with center at x. Then the volume of c(&) differs from jdet(c*jx)j by a term whose size is bounded by an x-dependent multiple of rnþ1. This is seen by writing a first order Taylor’s expansion for c near x, thus writing c(x þ y) ¼ c(x) þ c*jx·y þ rx(y) where jrx(y)j is bounded by an x-independent multiple of jyj2. The claim about the volume of c(&) follows from the fact that a

Appendix 14.2

Integration on manifolds

linear transformation multiplies volumes by the absolute value of its determinant. Part 3: Fix a finite, open cover, U, of M such that each set U  U comes with a coordinate chart map jU : U ! Rn that is a diffeomorphism onto an open subset. Let {wU}U2U denote a subordinate partition of unity. See Appendix 1.2 if you are hazy about such things. Let O denote a nowhere zero n-form that defines the given orientation for TM. If U 2 U, write jU* O as sU dx1 ∧ . . . ∧dxn with (x1, . . . , xn) denoting the Euclidean coordinates for Rn. Here, sU is a nowhere zero function on the image of jU. Let eU denote the sign of sU. Ð Let X denote any given n-form. The integral of X over X is denoted by M X and it is defined as follows: If U 2 U, write jU*X as zU dx1 ∧ . . . ∧ dxN with zU a function on the image of jU. Granted this notation, the integral of X is ð n ð**Þ ∑U2u eU ððj1 U Þ*wU ÞzU d x: Rn

Part 4: What follows explains why this definition associates to any given n-form a number that is independent of the chosen coordinate chart cover of M. Suppose that U0 is a second coordinate chart cover. There is a version of the integral above for U0 , this defined by a partition of unity {w0U0 }U0 2 U0 . I have to show that ð ð n n 1 * *0 0 0 0 0 ∑U2 u eU ððjU Þ wU ÞzU d x ¼ ∑U 2u eU ððj1 U0 Þ w U0 ÞzU d x: Rn

Rn

Using the fact that U2U wU ¼ 1 and U0 2 U0 w0U0 ¼ 1, write the respective leftand right-hand expressions above as ð n 0 0 ∑U2u;U 2u0 eU ððj1 U Þ*ðwU w U0 ÞÞzU d x and ð Rn n 0 0 ∑U2u; U0 2u0 eU0 ððj1 U0 Þ*ðwU w U0 ÞÞzU d x: Rn

This understood, it is sufficient to verify the following: Fix any given U0 6¼ . Then the corresponding integrals, (U, U0 ) 2 U  U0 with U ð ð *ðw w0 0 ÞÞzU dn x and e 0 ððj10 Þ*ðw w0 0 ÞÞz 0 dn x; eU ððj1 Þ U U U U U U U U ¨

Rn

Rn

agree. To see that this is the case, introduce the coordinate transition function n c ¼ jU  (j1 U0 ). The latter is a diffeomorphism that maps the region in R where the right-hand integrand is nonzero to that where the left-hand integrand is nonzero. I use det(c*) to denote the determinant of the differential of c, this a nowhere zero function on the domain of c. These are relevant by virtue of the fact that zU0 ¼ det(c*)c*zU and eU0 ¼ sign(det(c*))eU. Meanwhile,

195

196

14 : Curvature polynomials and characteristic classes 1 0 0 0 (j1 U0 )*(wUw U0 ) ¼ c*((jU0 )*(wUw U )). This understood, let f denote the integrand on the left-hand side above. The integrand on the right-hand side above is the function jdet(c*)j c*f. Granted this last observation, then the desired equality between the integrals above follows from the change of variables property given in Part 2 above. Part 5: Consider the integral of n-form that can be written as X ¼ f O where O is a nowhere zero n-form that defines the given orientation, and where f is a nonnegative function that is greater than zero at some point. To see that the integral of X is nonzero, it is enough to note the following: Let U denote a given coordinate chart from the cover U that is used in Part 3 to compute the integral of X, and let jU : U ! Rn denote the associated embedding. Then the function zU that appears in U’s contribution to the integral of X is equal to eU ((j1 U )*f ). This the case, then the claimed positivity follows from the positivity property stated in Part 2. Suppose instead that X ¼ da where a is an (n 1)-form. Let U denote a given coordinate chart cover that is used in Part 3 to define the integral of X. Write the (n  1)-form a as U2 U d(wUa). This done, it follows that the integral of X can be written as ð X dðwU aÞ U2 u

U

because the form d(wUa) is nonzero only in U. As such, its pull-back to Rn via j1 U has support in a bounded set. Write this pull-back using the standard coordinate basis for the (n  1)-forms on Rn as aU ¼ v1 dx2 ∧ ∧ dxn  v2 dx1 ∧ dx3 ∧ ∧ dxn þ þ ð1Þn1 vn dx1 ∧ ∧ dxn1 : P

@vi Having done this, it follows that dðwU aÞ ¼ 1in @x . Granted the latter, then i the vanishing of the integral of da follows from the divergence property from Part 2.

The example of Sn: The examples in Chapter 14.7 involve integrals of n-forms on the sphere Sn. What follows computes a few. To set the stage, view Sn as the set of points x ¼ (x1, . . . , xnþ1) 2 Rnþ1 with jxj ¼ 1. The form O ¼ x1 dx2 ∧ ∧ dxn  x2 dx1 ∧ dx3 ∧ ∧ dxnþ1 þ þ ð1Þn xnþ1 dx1 ∧ ∧ dxn

Appendix 14.3

The degree of a map

on Rn restricts to TSn as a nowhere zero n-form. To see this, remark that the xnþ1 > 0 part of Sn has a coordinate chart whose inverse is the map from interior of the ball in Rn that sends   U ¼ ðU1 ; : : :; Un Þ ! U1 ; : : :; Un ; ð1  jUj2 Þ1=2 : The pull-back of O using this map is the n-form  1=2 1  jUj2 du1 ∧ ∧ dun : Thus, O is nonzero where xnþ1 > 0. A similar calculation finds it nonzero for xnþ1 > 0. You can check points on the equator by repeating the previous observations with xnþ1 replaced by the other coordinate functions. In the case of S2, the form O can be written using the spherical coordinates (y, j) as O ¼ siny dy ∧ dj. You can either use the latter formula, or the representation above as (1  juj2)1/2 du1 ∧ du2 to see that its integral is 4p.

Appendix 14.3

The degree of a map

This appendix first proves the proposition in Chapter 14.7 that defines the degree of a map. The appendix then computes the degree for a map from a certain surface in R3 to S2 of the sort that is described by Example 14.6. What follows is Proposition 14.7 from Chapter 14.7. Proposition 14.7 Suppose that X is a compact, oriented, n-dimensional manifold and suppose that M is another such manifold  Suppose that c: M ! X is a smooth map. There exists an integer p with the Ð Ð following property: Let o denote a n-form on X. Then c*o ¼ p o . This M X integer p is, by definition, the degree of c.  Homotopic maps from M to X have the same degree. Proof of Proposition 14.7 The assertion in the second bullet is proved first. To start this proof, remark that maps c0 and c1 from M to X are homotopic if there exists a smooth map from [0, 1]  M to X that equals c0 on some neighborhood of {0}  M and is equal to c1 on some neighborhood of {1}  M. Meanwhile, as o is an n-form and X is n-dimensional, it follows that do ¼ 0. This being the case, Chapter 12.2 explains why c1*o  c0*o is the exterior derivative of an (n 1)-form on M. It follows as a consequence that c1*o and c0*o have the same integral. Turn now to the proof of the claim made by the first bullet. The proof has six steps.

197

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Step 1: The proof benefits from a review of something said earlier. To start, suppose that Y is a smooth, oriented n-dimensional Riemannian manifold. Let y 2 Y denote a given point. Chapter 9.3 describes a certain sort of coordinate chart map that is defined on a small radius, metric ball centered on y. These are the Gaussian coordinates. Let U denote a ball on which these coordinates are defined, and let j: U ! Rn denote the corresponding coordinate chart map. As explained by the proposition in Chapter 9.3, the metric on Y pulls back via j1 to a Riemannian metric on the image of j in Rn that is given by a symmetric, positive definite n  n matrix valued function whose (i, j) entry has the form gij ¼ dij þ Kij(x) where jKijj  c0 jxj2 with c0 denoting a constant. Here, dij is 1 if i ¼ j and 0 otherwise. The metric also determines a preferred section of ∧n T*Y, this I denote by OY. It is the unique, unit normed section that defines the given orientation. If {e1, . . . , en} is an oriented, orthonormal frame for T*M at a given point, then OY ¼ e1 ∧ . . . ∧ en. It follows from this that OY pulls back via the inverse of the Gaussian coordinate chart map as an n-form that can be written as ðj1 Þ*OY ¼ ð1 þ kÞdx1 ∧ ∧ dxn where jkj  c0 jxj2. Here, c0 is another constant. If O  Y is an open set, then the volume of O is, by definition, the integral of OY over O. If O is a ball of radius r centered on y, then it follows as from the depiction  n just given of OY that the volume of O can be written 2 1 m where m depends only on n, and where jzj  c r . Here 1 þ zðrÞ r n 0 nþ1 n again, c0 is independent of r. Step 2: It proves useful to invoke a special case of Sard’s theorem (Theorem 1.3; Chapter 1 states the full version). Fix a Riemannian metric on X to measure distances and norms. Use OX to denote the unique n-form on X with norm 1 that defines the orientation. Let CX  X denote the set of points with the following property: If x 2 CX, then c1(x) contains a point where the determinant of c’s differential is zero. A point in CX is said to be a critical value of c. Theorem 14.14 (the n-to-n Sard’s theorem) Given e > 0, there exists an open Ð set Ve  X that contains CX and is such that Ve OX is less than e. Proof of Theorem 14.14 Fix also a Riemannian metric on M to measure distances and norms. Let p denote a given point in M and let U  M denote a Gaussian coordinate chart centered at p. Introduce j: U ! Rn to denote a corresponding Gaussian coordinate chart map that send p to the origin also. Let x 2 c(p) and let U0  X denote a Gaussian coordinate chart centered at x. Introduce j0 to denote a Gaussian coordinate chart embedding from U0 onto an open set in Rn that sends x to the origin. Use cU0 ,U to denote the composition of j0  c  j1, this mapping some open neighborhood of 0 2 Rn into another

Appendix 14.3

The degree of a map

neighborhood of 0 in Rn. Fix r > 0 so that the ball of radius r centered at the origin in Rn is contained in the domain of cU0 ,U. As explained momentarily, the image of this ball via cU,U0 is contained in an open set, Or , whose volume obeys ð*Þ

volume ¼ðOr Þ  cðlp þ rÞrn

where lp is the absolute value of the smallest eigenvalue of the differential of cU,U at the origin in Rn. Meanwhile, c denotes a number that is independent of r and p 2 M. Accept this claim for the moment to complete the proof of the n-to-n Sard’s theorem. To complete the proof, choose r > 0 but small enough so that the following is true: Fix any point p 2 M. A ball of radius r centered at p is contained in a Gaussian coordinate chart centered at p, and the image via cr of this ball is contained in a Gaussian coordinate chart centered at c(p). Note that this is the case for all sufficiently small r by virtue of the fact that M is compact and c is continuous. Now, let CM denote the set of critical points of c. These are the points where c’s differential is not an isomorphism. Let Wr  CM denote a maximal set of points in CM with the property that distinct points in Wr have distance at least 14 r apart. Let Nr denote the number of elements in Wr. This set has at most c1rn members as can be seen by the following argument: The set of 1 balls in M with centers on the points in Wr and radius 16 r are disjoint. Let OM denote the n-form with norm 1 that defines the orientation. It follows from n what is said in Step 1 that the volume of each such ball is no less than c1 1 r where c1  1 is a constant that is independent of the point in question or r. Granted this last fact, it follows that volume of the union of the radius 14 r balls n centered at the points of Wr is at least c1 1 r Nr. This must be less than the n volume of M. Thus, Nr  c1r . Granted the latter bound, remark next that the union of the balls of radius r centered on the points in Wr covers CM. Thus, the image of this union via c covers CX. It follows from (*) that the image of these balls is contained in an open set on which the integral of OX is bounded by c rnþ1Nr, and thus by cc1r. Turn now to the proof of (*). View the differential of cU0 U at the origin as a linear transformation from the vector space Rn to itself. Let A denote this linear transformation. I can find orthonormal bases {e1, . . . , en} and {e0 1, . . . , e0 n} for Rn so that A acts as an upper triangular matrix with (n, n) entry equal to the absolute value of A’s smallest eigenvalue. Thus Aek ¼ jk Akje0 j and An,n ¼ lp with lp as in (*). This has the following implication: Fix r > 0 and let &r denote the n-dimensional cube where each coordinate has absolute value less than r. Then A maps &r in to an n-dimensional rectangle where the coordinates (x1, . . . , xn) are constrained as follows: If k < n, then jxkj is no greater (n þ 1  k)jAj r and jxnj  lpr. As a consequence, the volume of A(&r) is less than the volume of this cube, hence less than c0 lp rn with c0 independent of r and the point in p.

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Granted the preceding, use Taylor’s theorem with remainder to write cU0 ,U near the origin as the map x ! Ax þ e where jej  c0 r with c0 independent of r and p. Given what was just said about rectangles, this implies that cU0 ,U maps the ball of radius r centered at the origin in Rn into the rectangle where the coordinates (x1, . . . , xk) are constrained so that jxkj  (n þ 1  k)(jAj þ c1r) r for k < n and jxnj < c1 (lp þ r) r. Here, c1 is independent of r and of p. This implies what is asserted in (*). Step 3: This step asserts and then proves an analog of the n-to-n Sard’s theorem, this a bound for the integral of c*OX over the set c1(Ve). Ð Lemma 14.15 Given d > 0, there exists e such that c1 ðVe Þc*OX < d: Proof of Lemma 14.15 Let x 2 X and let U0  X denote a Gaussian coordinate chart centered at p. Introduce j0 to denote the Gaussian coordinate chart embedding from U onto an open set in Rn. Write ((j0 )1)*OX as z dx1 ∧ . . . ∧ dxn with z a strictly positive function on the image of j. Suppose p 2 c1(x). Fix a corresponding Gaussian coordinate chart U  M containing p, and let j: U ! Rn denote the associated embedding. Use cU0 ,U to denote the composition of j0  c  j1: j(U0 c1(U0 )) ! j0 (U0 ). It follows from what is said in Chapter 5.5 that (j1)*c*OX has the form ¨

200

detðcU0 ;U* ÞcU;U0 *ðzÞdx1 ∧ : : : ∧dxn where (cU0 ,U) denotes the differential of cU,U0 and det(·) denotes the determi* nant of the indicated n  n matrix. Given r > 0, let Wr denote the set of points in M with distance less than r from CM. Since CM is the set of points where c* is not an isomorphism, the determinant of (cU0 ,U)* vanishes on the j image of CM. This and the preceeding formula for (j1)*c*OX imply that there is an r and p-independent constant, c, such that jc*Oj < cr at all points Wr . This last observation has the following consequence: Given d > 0, there exists rd such that if r < rd, then  1 ð    c*OX   d: 2 Wr

Keeping the latter inequality in mind, note next that each point in MWr has a neighborhood that is embedded by c. This follows from the inverse function theorem (Theorem 1.1) since c’s differential is invertible at each such point. Since Wr is open, so MWr is closed and compact. This being the case, there exists Nr such that MWr/2 contains a collection of at most Nr Gaussian coordinate chart maps that cover MWr and that are mapped in a 1–1 fashion by c as an embedding onto their image. Let Ur denote such a collection of sets. The formula for (j1)*c*OX tells us the following: If U  U, then

Appendix 14.3

ð

c*ðOX Þ ¼ eU

U

The degree of a map

ð OX

cðUÞ

where eU 2 {1} is the sign of the determinant of the differential of cU0 ,U. This last equality has the following consequence: Fix e > 0 so as to define Ve using the n-to-n Sard’s theorem. Then   ð     c*ðOX Þ  Nr e:    ¨

c1 ðVe Þ

ðMWR Þ

With the preceding understood, take e < 12 N1 r d to deduce what is asserted by the lemma. Step 4: Now let o denote a given n-form on X. What is said in Steps 2 and 3 lead to the following: Given d > 0, there exists e > 0 such that  Ð Ð     X o  XVe o  d;  Ð Ð     M c*o  Mc1 ðVe Þ c*o  d: Indeed, this follows by writing o as f OX with f : X ! R a smooth function. To proceed from here, use the fact that XVe is compact to find a finite collection U0e of Gaussian coordinate charts centered at points in XVe / 2 with the following properties: First, the collection of charts covers XVe . Second, if U0  U0e is any given chart, then c1(U0 ) consists of a finite, disjoint collection, UU0 , of open sets that are each contained in a Gaussian coordinate chart which is mapped by c as an embedding onto its image in X. Fix a subordinate partition of unity for the open cover of X consisting of the sets in U0e and the extra set Ve. If U0 2 U0e , I use wU0 to denote the corresponding partition function. P Note in this regard that U2Ue0 wU0 is equal to 1 on XVe . With U0e as just described, it follows now that ð ð c*o ¼ ∑U0 2 U0e ∑U2UU0 c*o: M¼c1 ðVe Þ

U

Write o again as f OX and invoke the formula for (j1)*c*OX in Step 3 to write the preceding formula as 1 0 ð ð C B c*o ¼ ∑U0 2 U0e @∑U2 UU0 eU wU0 oX A: U0

M¼c1 ðVe Þ

Granted this last equality, and the fact that d can be taken as small as desired, the proposition follows if the sum U2 UU does not depend on the chosen set U0 2 U0e. 0

201

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Step 5: To see about the sum in question, fix a point x 2 XVe and a Gaussian coordinate chart, U0 , centered at x. Let j0 denote the corresponding Gaussian coordinate chart map. Fix r > 0 but much smaller than the distance between x and Ve , and let B0  X denote the ball of radius r, centered at x. The map j0 identifies B0 with the ball of radius r about the origin in Rn. Now, fix a smooth function s on Rn that equals 1 on the ball of radius 12 r and equals zero on the complement of the ball of radius 34 r. Use this function to map B0 to Sn as follows: This map sends x 2 B0 to the point on the unit sphere in Rnþ1 with coordinate  1=2 s2 jxj2 þ ð1  2sÞ2 ðsx; 1  2sÞ 2 Rn  R: Denote this map by fx. Note that there is a small ball centered on the north pole in Sn whose inverse image via fx is a small ball centered on 0 in Rn. Use D  Sn to denote this ball about the north pole in Sn. Note also that a neighborhood of the boundary of the ball of radius r is mapped by fx to the south pole in Sn. The map fx can be used to construct a map, Fx , from X to Sn. The map Fx sends the complement of Bx0 to the south pole, and it sends a point p 2 B x0 to fx(j0 (p)). Let OSn denote an n-form on Sn that gives the orientation that is consistent with fx, and has norm 1 as measured with the round metric. Fix next a nonnegative function, u, on Sn which is nonzero on the complement of D, and is such that the integral of u OSn is equal to 1. Choose u so as to be invariant under rotations of Sn with axis the north pole. The change of variable formula for integration tells us that the integral over X of the n-form Fx*(u OSn) is equal to 1. This understood, it follows from what is said in Step 4 that the integral over M of c*(Fx*(u OSn)) is equal to U2 UU eU. Step 6: Recall that the second bullet of Proposition 14.7 as stated at the start of this appendix has already been proved. Granted this, and granted the conclusions of Step 5, the assertion that the sum U2 UU eU is independent of U0 follows if it can be shown that the maps Fx  c and Fx0  c are homotopic in the case that x and x0 are any two points in X. In particular, this will be the case if Fx and Fx0 are homotopic. Note, by the way, that the question of a homotopy between Fx and Fx0 makes no reference to either M or the map c. In any event, the fact that the maps Fx and Fx0 are homotopic follows from what is said by Proposition 8.4 about geodesics. To see how this comes about, choose a smooth path, say a geodesic, from x to x0 . Let g denote this geodesic. Fix an oriented, orthonormal frame at x, and then parallel transport the frame vectors using the Levi-Civita connection to obtain a trivialization of the orthonormal frame bundle of TX along g. Compose this frame with a smoothly varying, t-dependent rotation if needed so that the resulting 1-parameter family of frames gives the respective frames at x and x0 that are used to define the Gaussian coordinate charts and thus the maps Fx and Fx0 . Let {e1jg(t), . . . , enjg(t)} denote the frame at 0

0

Appendix 14.3

The degree of a map

the point g(t). Use this frame for g*T*X to define a map, W, from R  Rn into M as follows: The map sends a given point t 2 R and x 2 Rn to the point expg(t)(1inxieijg(t)). Proposition 8.4 tells us that this map W is smooth. Given that X is compact, there exists rX > 0 with the following significance: Let B0  Rn denote the ball of radius rM. Then the restriction of W to R  B0 embeds each t 2 R version of {t}  B0 so as to give the inverse map of a Gaussian coordinate chart map centered at g(t). Let {jt}t2R denote the corresponding, 1-parameter family of Gaussian coordinate chart maps. Then these constitute a smoothly varying family of maps, and this understood, it follows that the family {Fg(t)}t2R varies smoothly with t also. With the proposition stated at the outset of this appendix in hand, consider by way of a sample calculation, the proof of the following claim from Example 14.6: Fix g  0. There is a surface  in R3 of genus g with the property that the association to any given point of the corresponding outward pointing normal vector defines a degree 1  g map from the surface to S2. The surface I have in mind is the one that is described in Chapter 1.3 as the level set of a function f: R3 ! R that has the form f(x, y, z) ¼ z2  h(x, y) with h chosen to have the following properties:    

h(x, y)  0 where jxj2 þ jyj2  1. k 2 For each k ¼ {1, . . . , g}, require h(x, y)  0 where jx  2g j þ jyj2  r2 . h(x, y) > 0 otherwise. @ @ hdx þ @y hdy 6¼ 0 at any point where h ¼ 0, thus on the circle of dh ¼ @x radius 1 about the origin, and on the g circles of radius r about the respective k points in the set fðx ¼ 2g ; y ¼ 0Þg . k¼1; : : : ;g

1 . I now add g þ 1 extra conditions on the Here, r is positive, but less than 100g function h:



@2h @y2

< 0 at (1, 0, 0).



@ h @y2

k < 0 at fð2g  r; 0; 0Þg

2

1kg

.

These points comprise half of the set of points on the x-axis where the value of h makes the transition from negative to positive. To say more about the significance of these points, note that the image of a point (x, y, z) 2  via the map n is the unit vector !1=2    2  2  @h   2  @h  @h @h   þ   þ 4z n¼  ;  : ; 2z  @y   @x  @x @y It is a direct consequence of this depiction of the map n that the point (1, 0, 0) 2 S2 has precisely gþ 1 inverse images via n in , these being the k points given above, thus (1, 0, 0) and the points in the set fð2g  r; 0; 0Þg . 1kg The condition on the second derivatives implies that the differential of the map

203

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n is an isomorphism at each of these points. However, the sign of its determinant is positive at the point (1, 0, 0) and negative at the other points. In any event, the fact that the differential is an isomorphism has, via Chapter 1.1’s inverse function theorem, the following implication: There is a small radius disk centered at the point (1, 0, 0) in S2 whose inverse image via n has g þ 1 disjoint, inverse images, each mapped diffeomorphically by n onto the original disk. One of these g þ 1 inverse images contains (1, 0, 0) and the rest are k distinguished as follows: Each point from the set fð2g  r; 0; 0Þg is 1kg

contained in one and only one of these inverse images. I use D  S2 to denote this disk. Now, let O denote the standard area form on S2, this the constant section of 2 2 ∧ S that gives the orientation and has norm 1 as defined by the round metric. Fix a smooth function u: S2 ! [0, 1] that is nonzero only in D and is such that u O has integral equal to 1. Given that n maps each component of n1(D) diffeomorphically onto D, and given what was said in the preceding paragraph about the sign of the determinant of the differential of n at (1, 0, 0) and the k  r; 0; 0Þg1kg , it follows from the change of points that comprise the set fð2g variable formula for the integral as given in Appendix 14.1 that the integral of n*(u O) over  is 1  g. Since the integral of u O is 1, it follows by definition that n has degree 1  g.

Additional reading  Characteristic Classes, John Milnor, Princeton University Press, 1974.  From Calculus to Cohomology: De Rham Cohomology and Characteristic Classes, I. H. Madsen and Tornehave Jxrgen, Cambridge University Press, 1977.  Curvature and Characteristic Classes, J. L. Dupont, Springer, 1978.  The Geometry of Differential Forms, Shigeyuki Morita, Teruko Nagase and Katsumi Nomizu, American Mathematical Society, 2001.  Differential Forms in Algebraic Topology, Raoul Bott and Loring W. Tu, Springer, 1982.  Differential Topology, Victor Guillemin and Alan Pollack, Prentice-Hall, 1974.  An Introduction to Manifolds, Loring W. Tu, Springer, 2007.  Topology from the Differentiable Viewpoint, John W. Milnor, Princeton University Press, 1997.

15

Covariant derivatives and metrics

A very large part of differential geometry exploits various relations between certain sorts of covariant derivatives and metrics, especially metrics on the tangent bundle. This chapter introduces and discusses some of these fundamental relations.

15.1

Metric compatible covariant derivatives

Suppose that p: E ! M is a vector bundle with fiber Rn. Assume that E also has a fiber metric. By definition, the metric is a positive definite, symmetric section N of the bundle Hom(E E; R). The inner product of vectors v, u 2 Ejx is denoted for the time being by hv, ui; and the norm of v 2 E is denoted jvj. Suppose that K ! M is some other vector bundle on M. The metric induces a vector bundle N N homomorphism from the bundle (E K) E to K; thus a section of N N Hom((E K) E; K). This homomorphism is defined first on the decomposN N able elements in (E K) E and it is then extended by linearity to the whole of N N (E K) E. The action of the homomorphism on a decomposable element N N (v k) u is given by hv, ui k. There is a corresponding section of N N N N Hom(E (E K); K), this sending v (u k) to hv, ui k. These metric induced N N N N elements in Hom((E K) E; K) and Hom(E (E K); K) are both also denoted for now by h , i. Let  now denote a covariant derivative on sections of E. The covariant derivative  is said to be metric compatible when the following is true: Let s, s0 denote any given pair of sections of E. Then hs, s0 i is a function on M, and its exterior derivative obeys dhs; s0 i ¼ hs; s0 i þ hs; s0 i: Note that taking s ¼ s0 implies that

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15 : Covariant derivatives and metrics

djsj2 ¼ 2hs; si: To see what this means in a less abstract setting, suppose that U  M is an open set and that {eb}1bn is a basis of sections for E over U. If  is any given covariant derivative, then its action on a basis vector eb can be written as eb ¼ ∑1an ea

N

aab

where each element from the set {aab}1a,bn is a 1-form on U. This the case, the n  n matrix of 1-forms a with the components {aab}1a,bn defines the covariant derivative on U. If the basis is orthonormal and if  is metric compatible, then aab ¼ aba, which is to say that a is a lie(SO(n))-valued matrix of 1-forms. What follows is another, equivalent way to state the metric compatibility condition. To start, suppose that K ! M is any vector bundle and  is a covariant derivative on K. A section, s, of K is said to be covariantly constant if s ¼ 0. A covariant derivative on E is metric compatible if and only if N N E; R) ¼ E* E*, the metric, when viewed as a section of K ¼ Hom(E is covariantly constant with respect to covariant derivative on sections of N E* E* that is induced by . By way of reminder, if  is any given covariant derivative for sections of E, then it induces one for sections of E*, and for sections of all multiple tensor powers and direct sum bundles whose factors are E and/or E*. This induced derivative can be defined in two equivalent ways. The first is to view  as coming from a connection on the principal bundle PGl(E) of frames for E, and then using the fact that E* and multiple tensor products whose factors are E and/or E* are associated vector bundles to PGl(E) using representations of the general linear group. This point of view is described in Chapter 11.5. An equivalent definition first defines  on sections of E* as follows: Let w denote any given section. If s is a section of E, then the evaluation of w on s is a function on M, this denoted by w·s. This understood, then w is defined by the requirement that dðw sÞ ¼ ðwÞ s þ w ðsÞ N

where w, a section of E* T*M, is viewed in the left-most term on the right side above as a section of Hom(E; T*M) and w in the right-most term is viewed N as a section of Hom(E T*M; T*M). Granted the latter definition of  on E*, the induced covariant derivative is defined on sections of multiple tensor powers by defining it first on decomposable sections, thus those of the form N N u2 . . . ; and then extending this definition by linearity. The definition u1 of  on a decomposable element is such that ðu1

N

u2

N

Þ ¼ u1

N

u2

N

þ u1

N

u2

N

þ etc:

15.1

Metric compatible covariant derivatives

Metric compatible covariant derivatives exist in all cases. In fact, any given such covariant derivative corresponds to a connection on the bundle PO(E) of orthonormal frames in E. Conversely, any connection on PO(E) defines a metric compatible covariant derivative. To see why this is, go to an open set U  M where there is an isomorphism jU : EjU ! U  Rn. As explained in Chapter 7.1, such a set U can be found with an isomorphism jU that gives a fiberwise an isometry with respect to the metric on E and the Euclidean metric on Rn. Assume in what follows that such is the case here for jU. Let s denote a section of E. Then jU(s) appears as the section x ! (x, sU(x)) of U  Rn where sU maps U to Rn. Meanwhile, jU(s) appears as the section x ! (x (s)U) of the N bundle (U  Rn) T*MjU where (sU) can be written as dsU þ aUsU with aU a Gl(n; R)-valued 1-form on U. Since jU is an isometry, the inner product of sections s and s0 on U gives the function sUT sU0 , and its exterior derivative is therefore dsTU s0U þ sTU ds0U : This can be written as ðsU ÞT s0U þ sTU ðs0 ÞU  sTU ðaTU þaU Þs0U : T

This implies that  is metric compatible if and only if aU ¼ aU. This is to say that aU is a lie(SO(n))-valued 1-form. If such is the case, then  comes from a connection on PO(E). What is essentially the same calculation shows that a connection on PO(E) supplies E with a metric compatible covariant derivative. Now suppose that  is a metric compatible connection, and let F denote its curvature 2-form, this a section of the associated bundle (PO(E) ad lie(SO(n))) N ∧2 T*M. Said in colloquial terms, F is a 2-form on M with values in the skew-symmetric endomorphisms of E. Note in this regard that the metric defines what is meant by skew-symmetric: An endomorphism m of E is skewsymmetric if and only if h · , m(·)i ¼ hm(·), · i. What follows is one additional observation in the case that E ¼ T*M with M oriented. A metric on TM induces one on T*M (and vice versa). Thus, in either case, M has a canonical, nowhere zero section of the real line bundle ∧n T*M, this the section with length 1 (as defined by the metric) and giving the chosen orientation. Here n ¼ dim(M). This section is denoted here by mM. To be more explicit, suppose that U  M is an open set and that {e1, . . . , en} is an orthonormal frame for T*M over U. Then this canonical section of mM of ∧dim(M) T*M is mM ¼ e1 ∧ . . . ∧ en. If  is a metric compatible connection on T*M, then this section is covariantly constant; thus mM ¼ 0. To see this, write eb ¼ P a ab as before. Then 1an e a ðe1 ∧ ∧ en Þ ¼ ðea ∧ e2 ∧ ∧ en Þ

N a1 a

þ ðe1 ∧ ea ∧ ∧ en Þ

N a2 a

þ ;

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and each term in this sum is zero. Indeed, ea ∧ e2 ∧ . . . ∧ en ¼ 0 unless a ¼ 1, but a11 ¼ 0 since a is skew symmetric. The remaining terms vanish for analogous reasons. The n-form mM is said to be the metric’s volume form. Appendix 14.2 explains how to integrate the form mM over any given open set in M with compact closure. This integral is in all cases a positive number. It is deemed the volume of the set in question. The integral of mM over M is guaranteed to be finite when M is compact. The value of the latter integral is said to be the volume of M. An analogous notion of metric compatibility exists for covariant derivatives on a complex vector bundle with a Hermitian metric. To elaborate, recall from Chapter 6.1 that a complex vector bundle, E, with fiber Cn can be viewed as a real vector bundle with fiber R2n and an almost complex structure, j. The real bundle is denoted by ER. The almost complex structure is a section of End(ER) that obeys j2 ¼ i. The fiber of E at any given x 2 M is by definition the span of the complex eigenvectors of j on ERjx with eigenvalue i. As discussed in Chapter 7.2, a Hermitian inner on E comes from a positive definite inner product, h , i on ER with the property that hv, jui ¼ hjv, ui. This induced Hermitian inner product for E is denoted also by h , i. A covariant derivative for sections of E comes from a covariant derivative for sections of ER with the following property: Let s denote any given section of ER. Then (j s) ¼ js. Such a covariant derivative on sections of ER will give a metric compatible covariant derivative on sections of E if and only if it is a metric compatible covariant derivative on sections of ER. Metric compatible covariant derivates for sections of E are in 1–1 correspondence with connections on the principal U(n)-bundle PU(E) of unitary frames for E. N The curvature, F, in this case defines a section of (PU(E) ad lie(U(n))) ∧2 T*M, which is to say that it is a 2-form on M with values in the antiHermitian endomorphisms of E.

15.2

Torsion free covariant derivatives on T*M

As always, let M denote a smooth manifold. The exterior derivative gives a completely canonical way to obtain a section of ∧2 T*M from one of T*M. However, if T*M has a covariant derivative, then the exterior covariant derivative as defined in Chapter 12.4 gives a second way to obtain such a section. By way of a reminder, suppose that  is a covariant derivative on sections of T*M and that a is section of T*M, thus a 1-form on M. Then a is a section N of T*M T*M and antisymmetrizing after multiplying by 1 gives a section of ∧2 T*M. This section of ∧2 T*M is also denoted by d a.

15.2

Torsion free covariant derivatives on T*M

The torsion of the covariant derivative  as applied to the 1-form a measures the extent to which d a and d a differ: d a  da. What follows is an important observation: The two form d a  da can be written as d a  d a ¼ T a where T is a section of the bundle Hom(T*M; ∧2 T*M). This is to say that T takes no derivatives. Here is why: Suppose that f is a smooth function. Then dðfaÞ ¼ df ∧ a þ fda and d ðfaÞ ¼ df ∧ a þ fd a : Thus T(fa) ¼ f T a and this can happen only if no derivatives are involved. Indeed, this follows as an application of Lemma 11.1 with E ¼ T*M, E0 ¼ ∧2 T*M and with L ¼ d  d. A covariant derivative on T*M is said to be torsion free when T ¼ 0. If T ¼ 0, then d and d agree not just on 1-forms, but on p-forms for all p. This follows directly from the Leibnitz rule that both obey, namely d(a ∧ b) ¼ da ∧ b þ (1)deg (a) a ∧ db and likewise d(a ∧ b) ¼ da ∧ b þ (1)deg(a) a ∧ db. There exist in all cases, torsion free covariant derivatives. To construct such a thing, first fix a covariant derivative . The task is to find a section, a, of the N 0 ¼  þ a is torsion free. This is done by bundle End(T*M) T*M such that N first writing the bundles End(T*M) T*M and Hom(T*M; ∧2 T*M) in an alternate form so as to view a and T as sections of the same vector bundle. Once this is done, it is a simple matter to choose a so that d0 has zero torsion. N To start, use the identification of End(T*M) with the bundle T*M TM to N N TM) T*M. Meanwhile T is a view a as a section of the bundle (T*M section of Hom(T*M; ∧2 T*M) which is equivalent to being a section of N TM. As such, T can also be viewed as a section of (∧2 T*M) N N (T*M T*M) TM, albeit one that is antisymmetric with respect to the endomorphism that interchanges the two T*M factors. To continue, note that N N N N the bundle (T*M T*M) TM is isomorphic to (T*M TM)N T*M. N The desired isomorphism takes any given decomposable element (u u0 ) v N N to the element (u v) u0 . Granted all of these identifications, then a and T are seen as sections of the same vector bundle. Viewed in this way take a to equal 1 times T to obtain a torsion free covariant derivative. Repeat the construction just given for the case T ¼ 0 to verify the following basic fact: Suppose that  is a torsion free covariant derivative on T*M. N N Then so is  þ a if and only if a is a section of (T*M TM) T*M that is N N TM) T*M that symmetric with respect to the automorphism of (T*M interchanges the T*M factors.

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15.3

The Levi-Civita connection/covariant derivative

Suppose now that M has a Riemannian metric. Then its orthonormal frame bundle has a very special connection, this being the Levi-Civita connection. This is the connection described in Theorem 15.1 (The Levi-Civita theorem) There is one and only one connection on the orthonormal frame bundle that induces a torsion free covariant derivative on T*M. Said differently, there is one and only one covariant derivative for sections of T*M that is both metric compatible and torsion free. What follows are some comments to put this theorem in perspective. First, the bundle T*M is an associated vector bundle to the bundle of frames in TM. This said, a connection on this principal bundle induces a covariant derivative on the space of sections of T*M. The bundle T*M also inherits a canonical fiber metric from the metric on TM. There are any number of ways to see this. One way to see this is to write T*M as the just described associated bundle to the bundle, PGl(M), of frames in TM. As such, it is given as PGl(M) r Rn, where r is the representation of Gl(n; R) on Rn that has a matrix g act so as to send a vector v to (g1)T. Use the given metric to define the principal bundle PO(n) of orthonormal frames in TM. This identifies PGl(M) as the principal bundle given by PO(n) def Gl(n; R) where def is the defining representation of O(n) in Gl(n; R). This identifies T*M with TM because the representations of O(n) on Rn that have g act on a vector v 2 Rn to give gv and (g1)Tv are one and the same representations of O(n). This identification gives T*M the fiber metric that the theorem is referring to. Here is another way to see this N identification between TM and T*M: The metric is a section of Hom(TM TM; R) and this bundle is canonically isomorphic to Hom(TM; T*M). Viewed in this light, the metric defines an isomorphism from TM to T*M. The induced metric on T*M is defined by the requirement that the latter isomorphism define an isometry. Proof of the Levi-Civita theorem (Theorem 15.1) I find the only proof I know long on index manipulation and short on intuition. In any event, choose an orthornormal frame for T*M over some coordinate chart U  M. Use {e1, . . . , en} to denote the chosen frame. Then ei can be written as ð*Þ

ej ¼ ei

N

ajik ek

where repeated indices come implicitly with a summation over the indexing set {1, . . . , n}. This covariant derivative is metric compatible if and only if akij ¼ ajik for all 1  i  j  n:

15.4

A formula for the Levi-Civita connection

Meanwhile, write dej ¼ j

1 j i w e ∧ ek ; 2 ik

j

where w ik ¼ w ik for all indices 1  i, k  n. Thus,  is torsion free if and only if ij

kj

j

ak  ai ¼ w ik : These conditions on a are satisfied by 1 aijk ¼  ðw ijk  w jik þ w kji Þ: 2 To see that this is the only solution, remark that if there were two, then their difference, akij, would have to obey both akij ¼ akji and akij ¼ akj i . This is possible if and only if all akij are zero. To see why, call the first equality the antisymmetry condition and the second the symmetry condition. Then note that alternating applications of first the symmetry condition and then the antisymmetry condition lead to the following sequence of equalities jk ji ij ik ki akij ¼ akj i ¼ ai ¼ aj ¼ aj ¼ ak ¼ ak :

As the first and last terms are identical but for the sign, so each term must be zero.

15.4

A formula for the Levi-Civita connection

Equation (*) above says all that is needed to define the covariant derivative since it tells us how to take covariant derivatives of any given basis of orthonormal sections of T*M that are defined over an open set in M. Indeed, given such a basis {ei}1in, their covariant derivatives are defined by the rule ej ¼ ei

N

ij

ak ek

and if w ¼ siei is any other section of T*M, then w ¼ ei

N

ij

dsi þ si ak ej

N

ek :

Here as before (and in what follows), repeated indices come with an implicit summation over the indexing set {1, . . . , n}. To take the covariant derivative of a vector field, introduce the dual basis {va} of T*M, this defined so that the pairing is ei·va ¼ 1 or 0 depending on whether i ¼ a or i 6¼ a respectively. Then aa is defined so that

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0 ¼ dðei va Þ ¼ ei va þ ei va : This finds va ¼ aia k vi

N

k

e :

Suppose now that U is an open set in M and that c: U ! Rm is a coordinate chart. Here, I use m for the dimension of M. The Riemannian metric pulls back via c as the section gij dxi of T*Rm

N

N

dxj

T*Rm.

Proposition 15.2 The Levi-Civita covariant derivative  is given by N i dxi ¼ Gjk dxj dxk where Gijk ¼ 12 gin ð@j gnk þ @k gnj  @n gjk Þ: i i ¼ Gkj , it follows that d(dxi) ¼ Proof of Proposition 15.2 Noting that Gjk i j k Gjk dx ∧ dx ¼ 0. This verifies that  is torsion free. Consider now N (gnp dxn dxp). If this is zero, then  is also metric compatible. Write out this expression to see that such is the case:

ðgnp dxn

N

dxp Þ ¼ ð@ k gnp  grp Grnk  gnr Grpk Þðdxn

The coefficient of (dxn

N

dxp)

N

N

dxp Þ

N

dxk :

dxk on the right-hand side is

1 1 @ k gnp  ð@ n gpk þ @ k gpn  @p gnk Þ  ð@p gnk þ @ k gnp  @n gpk Þ: 2 2 This sum is zero. This being the case, the Levi-Civita covariant derivative has the form asserted by the proposition. Recall from Chapter 8.2 that this same Gijk appears in the definition of a geodesic. Indeed, the geodesic equation from Chapter 8.2 is the assertion that a map t ! g(t) from an interval I  R is a geodesic if and only if g is covariantly constant along g. This term has the following meaning: If g is viewed as a section over I of g*M, then its covariant derivative as defined using the pullback of  is zero.

15.5

Covariantly constant sections

As an aside, note that the term covariantly constant is used in greater generality. To elaborate, suppose that E ! M is a vector bundle and  is a connection on E. Recall from Chapter 15.1 that a section, s, of E is said to be covariantly

15.5

Covariantly constant sections

constant if s ¼ 0. Of course, if s ¼ 0, then ds must also vanish, and as ds ¼ F s, this requires Fs to vanish: The curvature 2-form, a section of N End(E) ∧2 T*M, must annihilate s. This, in general, is fairly strong constraint on the curvature, for it asserts that if v and u are any two vector fields on M, then the endomorphism F(u, v) of E annihilates s. If the span of the linear space of covariantly constant sections of E is constant over an open set in M, then this span defines a subbundle, E0  E. The covariant derivative acts as zero on sections of E0 , and so it induces a covariant derivative of sections of E0 with zero curvature. This gives a flat connection on the bundle PGL(E0 ). Now suppose that I  R is an interval and that g: I ! M a smooth map, the image is a path in M. A section s of g*E is said to be parallel transported along g, or just parallel along g, when s is covariantly constant with respect to the pull-back covariant derivative, g*. This is to say that (g*)s ¼ 0. To see what this means, fix any given isomorphism j: g*E ! I  V where V is Rn or Cn as the case may. Write j(s) as the section t ! (t, sj(t)), where sj maps I to V. Likewise, write j((g*)s) as the section t ! (t, ((g*)s)jdt) N T*I over the interval. In general ((g*)s)j has the of the bundle (I  V) d form dt sj þ aj sj where aj is a map from I to Gl(V). The section s is covariantly constant (equivalent terms are ‘parallel’ and ‘parallel transported’) when d sj þ aj sj ¼ 0: dt Granted this, it follows from what is said in Chapter 8.3’s vector field theorem about solving systems of ordinary differential equations that there exists, in all cases, a covariantly constant basis of sections of g*E. Suppose that {s1, . . . , sn} is such a basis. Define a new isomorphism from g*E to I  V by declaring that its inverse be given by the map that sends a given point P (t, (v1, . . . , vn)) to the point 1kn vkskjt. Let c denote this basis. It then follows by virtue of the fact that (g*)sk ¼ 0 for all k that the ððg*ÞsÞc ¼ dtd sj . Indeed, this is because s when written in terms of the basis {sk} if it has the form P k vk(·) sk where each vk(·) is a function on I. Then X    P N d ðg*Þs ¼ v sk dt þ k vk ðg*Þsk : k dt k Thus, the trivialization of g*E given by the covariantly constant basis makes g* look just like the ordinary derivative. This is a reason to prefer a basis of covariantly constant sections.

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15.6

An example of the Levi-Civita connection

Suppose that M is an n-dimensional submanifold in RN for some N  n. This makes TM into a subbundle of the product bundle M  RN. The Euclidean metric on RN supplies a Riemannian metric for TM. To be explicit, fix any point x 2 M. Then the inner product of vectors v and v0 in TMjx is their Euclidean inner product as vectors in RN that happen to lie in the subspace TMjx. Use g in what follows to denote this metric. The identification of TM as a subbundle of M  RN also gives a covariant derivative for sections of TM. To see how this comes about, let P: M  RN ! M  RN denote the bundle homomorphism that is given at each x 2 M by the orthogonal projection onto the vector subspace TMjx  RN. Write a section, v, of the product bundle M  RN as x ! (x, v(x)) where v: M ! RN is a smooth map. A section of TM (i.e. a vector field on M) is such a section x ! v(x) ¼ (x, v(x)) such that v: M ! RN lands in TMjx at each x. Thus, Pv ¼ v for a section of TM. N The bundle TM T*M can likewise be viewed as a subbundle inside the N bundle (M  RN) T*M. This is because it is spanned by the decomposable N elements of the form v a where a is a 1-form and v(x) ¼ (x, v(x)) with v: M ! Rn is such that Pv ¼ v at each x. If x ! v(x) ¼ (x, v(x)) is a section of M  RN (whether or not v(x) 2 TMjx), then x ! (x, dvjx) defines a section of N (M  RN) T*M. Here, dv is the vector of 1-forms that is obtained by taking the exterior derivative of each component of the vector valued map v. This understood, suppose that x ! v(x) ¼ (x, v(x)) defines a section of TM. Set v to be the N section of TM T*M given by x ! (x, Pdvjx). Here, P is defined on sections of N the bundle (M  RN) T*M by its action on the decomposable elements. The covariant derivative just defined is the Levi-Civita covariant derivative for the metric just defined. To see why this is, it is enough to check that this covariant derivative and the Levi-Civita covariant derivative have the same affect on any given vector field. Let ◊denote for the moment the Levi-Civita N covariant derivative. The difference,   ◊, is a section, a, of End(TM) T*M. The task is to prove that a is zero at each point. To do this for a given point x, the strategy is to choose a coordinate basis for M near x which makes the comparison relatively easy to do. Since a is a tensor, the assertion that ajx ¼ 0 does not depend on the chosen coordinate chart. To reiterate, the strategy for this proof exploits the fact that the assertion to be proved is coordinate independent. By the way, this same strategy should be considered first for any calculation of any coordinate independent statement in differential geometry. To start the argument, use the fact that the Euclidean metric on RN is translation and rotation invariant to make a translation and rotation of RN

15.6

An example of the Levi-Civita connection

so that x is the origin and so that the splitting RN ¼ Rn  RNn has the first factor tangent to M at x. Thus, TMjx is the factor Rn  Rn  RNn. The first n coordinates can serve now as coordinates on M near x. This is to say that M near x is the image of a neighborhood of the origin in Rn via a map, j, that sends y ¼ (y1, . . . , yn) to the N-tuple jðyÞ ¼ ðy1 ; . . . ; yn ; f1 ; . . . ; f Nn Þ where f ¼ ( f 1, . . . , f Nn): Rn ! RNn is such that f k ¼ 0 at y ¼ 0 and also   X @ df k ¼ f dya ¼ 0 at y ¼ 0: 1an @y k a The metric g appears in these coordinates as gabdyadyb where gabjy is the Euclidean inner product at j(y) 2 RN between j* @y@ and j* @y@ . Here, j* @y@ a b a is the push-forward to RN via the embedding j of the coordinate vector field @y@ : a

j*

X @ @ @f k @ ¼ þ : 1bNn @y @x @ya @xa nþk a

Thus, the metric coefficient gab is gab ¼ dab þ @ a f k@ b f k where dab ¼ 1 if a ¼ b and dab ¼ 0 otherwise. This formula can be used to compute the Levi-Civita covariant derivative of the coordinate 1-form basis {dya}1an. Recall that I use ◊ to denote the Levi-Civita covariant derivative. Then ◊dya can be written as ◊dya ¼ N Gabc dyb dyc where   1 ad @ @ @ a g þ g  g : Gbc ¼ g 2 @yb dc @yc db @yd bc I use in this last equation and subsequently the convention where {gab}1a,bn denotes the entries of the inverse of the matrix, g, with entries {gab}1a,bn; the position of the index as superscript or subscript is meant to convey the distinction between g and the inverse matrix g1. Here is the first place where the choice of coordinates is exploited: a The metric coefficient gab is the constant dab to order jyj2. Thus, Gbc ¼ O(jyj). a ◊ a In particular, Gbcjy=0 ¼ 0. This is to say that  dy ¼ 0 at y ¼ 0. As a consequence @ ¼ 0 at y ¼ 0 ◊ @ya N

dyc . To elaborate, this last equation follows from the because @y@ ¼ Gbac @y@ a b relation between the covariant derivatives on TM and T*M; they determine each other using the rule

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       d aðvÞ ¼ ◊ a v þ a ◊ v : Here, a is any given 1-form, v is any given vector field, and a(v) denotes the function that is obtained by pairing the 1-form with the vector field. The next task is to compute  @y@ at y ¼ 0. If this is also zero, then  ¼ ◊ as a claimed. To prove this equality, note first that the definition of  is such that    X @ @ @f k @ ¼P d þ  1hNn @y @x @ya @xa nþk a  2  X  N @ fk @ d dy ¼ P : 1hNn @y @y @x nþk d a Meanwhile, the projection P at any given point near x is the projection in RN onto the span at the given point of fj* @y@ g. In particular, it takes a vector a v ¼ vc @x@ c þ vkþn @x@nþk to    X X @ fk @ @ fk @ cb Pv ¼ g vb þ : v þ 1hNn k @y 1aNn @y @x @xc nþk b c What with these last two equations, it follows that   X X N @ @ f @2 fk @ @f k @ ¼ gcb 1hNn k þ dyd :  1hNn @y @x @yb @yd @ya @xc @ya nþk c Note in particular that the expression on the right-hand side vanishes at y ¼ 0 because the first derivatives of the functions f k all vanish at y ¼ 0. Thus, the covariant derivatives  and ◊ are indeed equal at x.

15.7

The curvature of the Levi-Civita connection

Suppose first that E ! M is a vector bundle with fiber Rn, that g is a fiber metric on E, and that  is a metric compatible connection on E. This means that g is a N covariantly constant section of the bundle E* E*, thus g ¼ 0. To see what this entails, suppose that U  M is an open set and {eb}1bn is an orthonormal basis of sections of E*. Thus, the metric g can be written as g ¼ eb

N b e :

Here and in what follows, repeated indices are implicitly summed over the N indexing set {1, . . . , n}. Meanwhile, eb can be written as eb ¼ ea aab where each a,b 2 {1, . . . , n} version of aab is a 1-form on U. The collection {aab}1a,bn can be viewed as the coefficients of a 1-form, a, on U with values in the vectors space of n  n antisymmetric matrices.

15.7

The curvature of the Levi-Civita connection

The condition dg ¼ 0 says that the curvature 2-form F annihilates N the section g. To understand what this implies, write (F)·eb ¼ ea (F)ab. The condition dg ¼ 0 says that (F)ab ¼ (F)ba; this asserts what was said previously in Chapter 15.1: The curvature of a metric compatible connection is a 2-form on M with values in the bundle of antisymmetric endomorphism of E. Now specialize to the case when E is the tangent bundle of M, and  is a torsion free connection. Let {eb} denote an orthonormal basis of 1-forms N ab N over some open set U in M. Write eb ¼ ea a and (F)eb ¼ ea (F)ab ab ab ac cb where (F) ¼ da þ a ∧ a . Write the 2-form indices of F explicitly: 1 c d ðF Þab ¼ ðF Þab cd e ∧ e 2 N

ab ab where (F)cd ¼ (F)dc . Given that d2 eb ¼ ea (F)ab, and given the torsion free condition, which is that d acts as d, it follows that (F)ab ∧ eb ¼ 0. This says that

c d b ðF Þab cd e ∧ e ∧e ¼ 0:

Here as before (and subsequently), repeated indices are implicitly summed over the indexing set {1, . . . , n}. What is written here says neither more nor less than ac ad ðF Þab cd þ ðF Þdb þ ðF Þbc ¼ 0: ab ¼ (F)ba If  is the Levi-Civita connection, then it is also the case that (F)cd cd, and this condition with the equation written above imply that

cd ðF Þab cd ¼ ðF Þab : ab is symmetric with respect to interchanging its first two This is to say that (F)cd and last two pairs of indices. Since it is antisymmetric in each pair, the curvature of the Levi-Civita connection can be viewed as a section of the bundle of symmetric endomorphisms of ∧2 T*M. Viewed in this way, F is called the curvature operator. The curvature of the Levi-Civita covariant derivative is called the Riemann curvature tensor. It is written with respect to a given orthonormal frame for ab T*M as (F)cd ¼ Rabcd. As noted above, it enjoys the following symmetries:

    

Rabcd ¼ Rbacd , Rabcd ¼ Rabdc , Rabcd ¼ Rcdab , Rabcd þ Radbc þ Racdb ¼ 0, e Rabcd þ c Rabde þ d Rabec ¼ 0.

The last one is the Bianchi identity dF ¼ 0. With regards to the notation of the last line, the Riemann tensor is viewed as the section

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Riem ¼

1 Rabcd ðea ∧ eb Þ 4

N

ðec ∧ ed Þ

N

of (∧2 T*M) (∧2 T*M), and, as such, it has the covariant derivative Riem, N N this a section of ((∧2 T*M) (∧2 T*M)) T*M which is written as Riem ¼

1 e Rabcd ððea ∧ eb Þ 4

N

ðec ∧ ed ÞÞ

N

ee :

With ea ¼ eb aab, this means e Rabcd ¼ @e Rabcd þ aaf Rfbcd þ abf Rafcd þ acf Rabfd þ adf Rabcf ; where @ e indicates the directional derivative of the function Rabcd in the direction of the vector field ee. The components of the Riemann curvature tensor are used to define a N symmetric section of T*M T*M, this being the Ricci tensor; its components are Ricac ¼ ∑b Rabcb : N a

ec. The trace of the Ricci tensor (its inner Thus, the tensor is Ricac e product with the metric) is called the scalar curvature: R ¼ ∑a Ricaa ¼ ∑ac Ricacac : The Bianchi identities imply in part that Ricci tensor obeys 1 e Ricac  c Ricae þ b Rabec ¼ 0 and c Ricac  a R ¼ 0: 2 The symmetric section of T*M

N

T*M with components

1 Ricab  Rdab 2 is called the Einstein tensor. Here dab is 1 if a ¼ b and 0 otherwise. The section N b N a N e ¼ ea e of T*M T*M is the Riemannian metric. The Ricci dab ea tensor, Einstein tensor and the scalar curvature play a central role in modern applications of metric geometry. The next chapter describes these tensors in various examples.

Additional reading  Riemannian Geometry: A Modern Introduction, Isaac Chavel, Cambridge University Press, 2006.

Additional reading

 Riemannian Geometry, Manfredo P. Carmo, Birka¨user, 1992.  Riemannian Manifolds: An Introduction to Curvature, John M. Lee, Springer, 1997.  Riemannian Geometry, Peter Petersen, Springer, 2006.  Riemannian Geometry, Sylvestre Gallot, Dominque Hulin and Jacques Lafontaine, Springer, 2004.

219

16

The Riemann curvature tensor

This chapter studies first some fundamental examples where the Riemann curvature tensor has an especially simple form. It then briefly discusses some of the ways in which the Riemann and Ricci curvatures are used to study questions in geometry and differential topology.

16.1

Spherical metrics, flat metrics and hyperbolic metrics

The examples that follow have, in some sense, the simplest possible Riemann curvature tensor. In these examples, the components of the Riemann tensor when written with respect to a given orthonormal frame are Rabcd ¼ kðdac dbd  dbc dad Þ: Here, k is a constant and dac is 1 if a ¼ c and 0 otherwise. Let {ea}1an¼dim(M) denote the orthonormal frame in question. The corresponding section Riem of N ∧2 T*M ∧2 T*M is Riem ¼

O 1 kðea ∧ eb Þ ðea ∧ eb Þ: 4

(Here, as in the previous chapter, repeated indices are summed.) Viewed as an endomorphism of ∧2 T*M, this section acts as k times the identity endomorphism. Let n denote the dimension of M. The corresponding Ricci tensor is such that Rac ¼ (n  1)kdac. Stated without reference to an orthonormal frame, this is saying that Ric ¼ ðn  1Þkg: Meanwhile, the scalar curvature is R ¼ n(n  1)k.

16.1

16.1.1

Spherical metrics, flat metrics and hyperbolic metrics

Spheres

Fix n  2, r > 0 and embed Sn in Rnþ1 as the space of vectors in Rnþ1 with length r. The corresponding metric obeys the equation above with k ¼ r2. What follows does the computation directly using the formula given in Chapter 8.4.3 for the metric of a submanifold in RN¼nþ1. Recall that the sphere of radius r is invariant under the SO(n þ 1) group of rotations of Rnþ1 about the origin. This understood, it is enough to verify the formula for R at the north pole, this where x1 ¼ x2 ¼ . . . ¼ xn ¼ 0 and xnþ1 ¼ r. The coordinate embedding in this case has f(y) ¼ (y1, . . . , yn, (r2  jyj2)1/2). The corresponding metric components {gab} are given by gab ¼ dab þ ðr2  jyj2 Þ1 ya yb : As the Christoffel symbols {Gabc} involve first derivatives of g and the curvature involves first derivatives of the Christoffel symbols, it is enough for the computation of Rabcd at y ¼ 0 to replace gab by its second-order Taylor’s approximation, this being g*ab ¼ dab þ r2 ya yb : Likewise, the Christoffel symbols can be replaced by their first-order Taylor’s approximations, thus by Gabc ¼ r2 dbc ya : The curvature 2-form at y ¼ 0 is Rab ¼ dGabc ∧ dyc ¼ r2 dya ∧ dyb. This is to ab say that Rcd is equal to r2 (dacdbd  daddbc).

16.1.2

Tori

Fix n  2 and fix a lattice   Rn, this being a set of basis vectors, {v1, . . . , vn}. Do not require any orthonormality. Define T to be Rn/Zn where an integer n-tuple (k1, . . . , kn) 2 Zn acts on x 2 Rn as the translation by the vector P 2p 1an kava. The quotient is a smooth manifold that is diffeomorphic to n S1. A convenient diffeomorphism is given as follows: Identify S1 as R / 2pZ. The diffeomorphism, j, sends a given point (t1, . . . ., tn) 2 R / 2pZ to P the equivalence class of the vector 1an ta va. As the Euclidean metric on Rn is translationally invariant, it descends to T to define a metric there. This is to say that the metric pulls back via f to the metric g ¼ gabdtadtb where gab ¼ va vb ;

221

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16 : The Riemann curvature tensor

with v·v0 denoting the Euclidean dot product. Since the Christoffel symbols involve a derivative of gab, they must all vanish; thus the metric so defined has Riemann curvature tensor equal to zero. Such a metric is said to be flat. Two lattices  and 0 give the same metric (up to a diffeomorphism of n S1) if there exists an integer-valued matrix m 2 Gl(n; R) with determinant 1 such that  ¼ m0 .

16.1.3

The hyperbolic space

Define R1,n to be R  Rn with the Lorentz metric l ¼ dt

O

dt þ dxa

O

dxa :

Here, I use t for the coordinate on the R factor and x ¼ (x1, . . . , xn) for the coordinates of the Rn factor. Although not positive definite, this bilinear form is nonetheless called a “metric”. The bilinear form is, however, nondegenerate. Like its spherical counterpart, this metric has a large symmetry group, known as the Lorentz group. This is the subgroup of Gl(n þ 1; R) of matrices m that preserve the quadratic form x2  t2. For example, it contains the SO(n) subgroup that fixes the t-coordinate. It also contains matrices that fix, say x2, . . . , xn and act on the (t, x1) coordinates as   coshðuÞ sinhðuÞ sinhðuÞ coshðuÞ with u 2 R. The subgroup generated by the matrices in SO(n) and the matrices of that fix all but one xa as just described is the subgroup, Lþ, of the Lorentz group that fixes the t  0 cone in R1  Rn. Fix r > 0 and let M  R1  Rn denote the manifold where x2  t2 ¼ r2 and t > 0. Although the Lorentz metric is not positive definite, the induced metric on M is positive definite. To see this, remark that a tangent vector v to M at a @ point (t, x) can be written as v ¼ vt @t þ va @x@ a where (vt , va) are such that tvt  xava ¼ 0. As t ¼ (r2 þ jxj2)1/2 > 0 on M, this means that vt ¼ (r2 þ jxj2)1/2 xava. Meanwhile, l(v, v) ¼ vava  vtvt, and as t > 0 on M, this tells us that lðv; vÞ ¼ va vb ðdab  ðr2 þ jxj2 Þ1 xa xb Þ: In particular, the eigenvalues of this quadratic form are 1 (with multiplicity n  1) and r2/(r2 þ jxj2), this last with multiplicity 1. The metric just described is called the hyperbolic metric. As is explained next, the Riemann curvature tensor for this metric is such that Rabcd ¼ r2(dacdbd  daddbc). The Ricci tensor has components Rab ¼ (n  1)r2 dab, this the negative, constant factor (n  1)r2 times the metric tensor. The scalar curvature is R is equal to the constant n(n  1)r2.

16.2

The Schwarzchild metric

To prove these assertions, use the fact that the Lþ acts to preserve both M and the Lorentz inner product l on R1,n on M to see that it is enough to prove this assertion at the point (t ¼ r, x ¼ 0) in M. To consider the curvature at this point, use the coordinates supplied by the embedding j: Rn ! M  R  Rn that sends y ¼ (y1, . . . , yn) to jðyÞ ¼ ððr2 þ jyj2 Þ1=2 ; y1 ; : : : ; yn Þ: The metric on M appears in these coordinates as gabdyadyb where gab ¼ lðj* @y@ ; j* @y@ Þ. Given that a

b

j*

@ @ @ ¼ ðr2 þ jyj2 Þ1=2 ya þ ; @ya @t @xa

it follows that gab ¼ dab  ðr2 þ jyj2 Þ1 ya yb : As in the spherical case, it is sufficient for computing Rabcd at y ¼ 0 to use the second-order Taylor’s approximation for gab and the first-order Taylor’s a approximation for Gbc . The subsequent calculations are identical to those done in the spherical case save for the factor of 1 between the two terms in the preceding expression for gab. This leads to the factor of 1 in the expression Rabcd ¼ r2(dacdbd  daddbc).

16.2

The Schwarzchild metric

What follows is an example of a metric with zero Ricci curvature, but a more complicated Riemann curvature tensor. This metric is called the Schwarzchild metric, and it is a metric on the noncompact manifold S2  R2. There is a 1-parameter family of such metrics, the parameter being m > 0. The metric is written first in a coordinate basis with the coordinates (t, r, y, j) where y and j are the usual angular coordinates on S2, where t is an R / (2pZ)-valued coordinate, and r > m. g ¼ ð1  m=rÞ4m2 dt þ r ðdy 2

O

N

O

dt þ ð1  m=rÞ1 dr 2

dy þ sin ydj

O

N

O

dr

djÞ:

Note in this regard that dy dy þ sin2y dj dj is the standard round metric on S2. The apparent singularity as r ! m is spurious and due to the fact that these coordinates are not valid on r ¼ m . Taking r ! m corresponds on

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S2  R2 to going to S2  {0}, as can be seen using the change of coordinates (t, r, y, f) with r ¼ (1  m/r)2 taking values in [0, 1). This change writes g as 4m2 ðr2 dt

O

dt þ ð1  r2 Þ4 dr 2 2

þ m ð1  r Þ ðdy 2

O

O

drÞ

dy þ sin2 ydj

O

djÞ:

The (r, t) part is a smooth metric on the interior of the radius 1 disk in R2, and the (y, f) part is a smooth, r-dependent metric on S2.

16.3

Curvature conditions

An oft-used strategy in differential geometry is that of finding a canonical metric on a given manifold. If the precise notion of “canonical” asserts a condition that is equivariant under the action of the diffeomorphism group, then the existence or not of a “canonical” metric says something about the smooth structure on M. Conditions that involve the Riemann curvature, the Ricci curvature, or the scalar curvature of the metric are often considered because these objects behave as one might hope under the action of the diffeomorphism group. To elaborate, suppose that g is a metric on M and f: M ! M is a diffeomorphism. View the Riemann curvature as a section of N (∧2 T*M) (∧2 T*M), view the Ricci curvature tensor as a section of N T*M T*M, and view the scalar curvature as a function on M. Then the Riemann curvature of f*g is the pull-back via f of the Riemann curvature of g. Likewise, the Ricci tensor of f*g is the pull-back via f of the Ricci tensor of g. And, the scalar curvature function of f is the pull-back of the scalar curvature function of g. These assertions follow from the three remarks that follow. Remark 16.1 Suppose that M and N are manifolds, that f: M ! N is a smooth map and that p: E ! N is a vector bundle with fiber either Rn or Cn. As explained in Chapter 5.1, the map f can be used to define pull-back vector bundle p: f*E ! M, this the subspace in M  E of points (x, v) with f(x) ¼ p(v). Let  denote a given covariant derivative on sections of E. Chapter 11.8 explains how f can be used to construct a corresponding covariant derivative on sections of f*E, this denoted by f*. Suppose now that P ! N is a principal bundle with fiber some group G and that E is associated to P via a representation r from G to some general linear group. Chapter 10.7 explains how the map f is used to define pull-back principal bundle f*P ! M. Let A denote a given connection on P. Chapter ^ 11.8 uses f to define the pull-back f*A on f*P. As explained in Chapter 11.5, the connection A defines the covariant derivate A on Sections of E. Meanwhile

16.3

Curvature conditions

^ f*A defines a covariant derivative on sections of f*A. You can unwind definitions to see that the latter is f*A. Here is one final point: Suppose that E has a fiber metric, m. Then m gives f*E a fiber metric, this denoted by f*m. To elaborate, view f*E  M  E as the subspace of pairs (p, v) such that f(p) ¼ p(v). Viewed in this way, the inner product as defined by f*m between pairs (x, v) and (x, v0 ) in f*E is m(v, v0 ). If a given covariant derivative  for sections of E is metric compatible, then the covariant derivative f* is metric compatible for the metric f*m. This is also checked by unwinding all of the various definitions. Remark 16.2 Specialize to the case when E ¼ T*N. As explained in Chapter 5.3, there is a canonical vector bundle homomorphism between the pull-back ^ for the latter moves the bundle f*(T*N) and T*M. Don’t confuse this with f, base point (in fact, over to N), while this canonical homomorphism does not. This bundle homomorphism is just the Chain rule in fancy dress. To elaborate, let x ¼ (x1, . . . , xn) denote coordinate functions on an open set in M centered on 0 given point p 2 M, and let y ¼ (y1, . . . , yn ) denote coordinate functions on an open set in N centered on f(p). Here, n ¼ dim(M) and n0 ¼ dim(N). The map f sends a given point near p with the coordinates x ¼ (x1, . . . , xn) to the point 0 0 near f(p) with coordinates given by y1 ¼ f1(x), . . . , yn ¼ fn (x). 1 n Meanwhile (dx , . . . , dx )jx is a basis for the fiber of T*M at the point with 0 coordinates x, while (dy1, . . . , dyn )jy¼f(x) is a basis the fiber of f*(T*N) at the image point. The aforementioned bundle map from f*(T*N)jx to T*Mjx acts with respect to this basis as dyk !

@fk a dx @xa

where repeated indices are summed as usual. Now suppose that m ¼ mikdyidyj is a metric on T*N. Then mikjf(·) dyidyk defines the metric f*m on f*(T*N). The homomorphism given by the precedN ing equation defines from m a symmetric section mf, of T*M T*M, this given by mik

@ji @fk a dx @xa @xb

O

dxb :

Note that this is nonnegative definite, but it won’t be an honest k a positive definite metric unless the homomorphism dyk ! @f from @xa dx f*(T*N) ! T*M is surjective. Suppose that  is a covariant for T*N. It is determined near f(p) by its action 0 on the basis of sections (dy1, . . . , dyn ). Write this action as i dyk ¼ Wik j dy

O

dyj :

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16 : The Riemann curvature tensor

The covariant derivative f* on f*(T*N) acts on sections of f*(T*N) to N give sections of f*(T*N) T*M. It’s action is defined by its action on the 0 sections (dy1, . . . , dyn )jf(·) near p by the rule ðf*Þðdyk Þ ¼

@fj ik i W dy @xa j

O

dxa :

Notice in particular that this formula does not give a covariant derivative on k a T*M unless there is an inverse to the homomorphism dyk ! @j @xa dx . Remark 16.3 Granted all of the above, suppose now that N ¼ M and that f is a diffeomorphism. If this is the case, then there is an inverse homomorphism, k @ca this given in the coordinate basis by dxa ¼ @yk jfðxÞ dy where c is the inverse to f. I can use the formula above now to define (f*)(dxa). This is done by first a viewing dxa ¼ @c dyk as a section of f*(T*N), then using the formula above, @yk k a and finally using the homomorphism dyk ! @f @xa dx to view the resulting covariant derivative as a section of T*M. Here is the final product:  2 a  @ c @ca ik  @fi @fj b O c ðf*Þðdxa Þ ¼ þ W dx dx :  @yi @yj @yk j fðxÞ @xb @xc To finish the story, remark first that this covariant derivative is compatible N @fi @fk with the metric mf ¼ mik @x dxa dxb . Moreover, if  is torsion free, then a @xb so is this covariant derivative. The point being that the condition that  is torsion is signified here by the fact that Wjik ¼ Wijk. Writing (f*)(dxa) as N a a dxb dxc, you see that Wbc obeys the corresponding symmetry condition; Wbc and so it is also torsion free. Granted the preceding, what follows are some common, diffeomorphisminvariant conditions that are imposed on the Riemann curvature, Ricci tensor, and scalar curvature. Vanishing Riemann curvature tensor: This is the condition that the Riemann tensor vanish everywhere on M. Thus, the Levi-Civita covariant derivative is defined by a flat connection. Positive or negative definite curvature operator: View the Riemann curvature N tensor as a symmetric section of (∧2 T*M) (∧2 T*M). As such, it defines a symmetric bilinear form on ∧2 TM. The condition here is that this form is either negative definite everywhere or positive definite everywhere. Positive or negative sectional curvature: View the Riemann curvature tensor again as a symmetric bilinear form Riem(·, ·) on ∧2 T*M. Let x 2 M and let u, v denote a pair of linearly independent vectors in TMjx. The sectional curvature of the plane spanned by u and v is, by definition, the number Riem(u ∧ v, u ∧ v). When the Riemann curvature is written in terms of an orthonormal basis {ea}

16.4

Manifolds of dimension 2: The Gauss–Bonnet formula N

for T*M as Riem ¼ 14 Rabcd ðea ∧eb Þ ðec ∧ed Þ, then the sectional curvature is Rabcduavbucvd where ua is the pairing of the vector u with the basis element ea, and likewise va is the pairing of v with ea. The positive or negative sectional curvature condition requires that all such sectional curvatures are respectively positive or negative. The Einstein equations: These equations ask the Ricci tensor to obey Ric ¼ 12 Rg where g is the metric, R is the scalar curvature. One can also ask that Ric ¼ 0, or that Ric be a positive or negative definite when viewed as a symmetric, bilinear form on TM. Constant scalar curvature: This demands that the scalar curvature function R be constant over M. More is said about some of these conditions in the subsequent parts of this chapter. There are other conditions that are special to certain dimensions (such as 4), or to metrics on manifolds with additional structure imposed such as an almost complex structure. The latter is defined in Chapter 17. (Chapter 17 also introduces a curvature condition that is special to this sort of manifold.)

16.4

Manifolds of dimension 2: The Gauss–Bonnet formula

Suppose that M is a 2-dimensional manifold with a Riemannian metric. There is but one independent component of the Riemann curvature on a 2-dimensional manifold, and this can be taken to be R, the scalar curvature. Indeed, fix an open set U  M where there is an orthonormal frame for T*M on U. Let {e1, e2} deote this frame. As defined with respect to this frame, the Riemann curvature tensor has R1212 ¼ 12 R and the other components of the Riemann tensor are either equal to R1212 or zero. The next proposition implies that the function R cannot be just any function on M. Proposition 16.4 If M is compact and oriented, then ð RmM ¼ 8pð1  genusðMÞÞ: M

Proof of Proposition 16.4 A Riemannian metric on an oriented surface endows the TM with an almost complex structure. Two observations are needed in order to define this almost complex structure. Here is the first: To say that M is orientable is to say that the line bundle ∧2 T*M admits a nowhere vanishing section. To say that M is oriented is to say that such a section has been chosen up to multiplication by a positive, nowhere zero function. The metric

227

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16 : The Riemann curvature tensor

on M gives such a section, that with norm 1 at each point. Let mM denote the latter. If {e1, e2} are an orthonormal basis for T*M at a given point, then mM at that point is e1 ∧ e2. Put mM away for the moment. Here is the second comment: The metric can be viewed as an isomorphism from TM to T*M. This is the homomorphism that sends any given vector v to the section of T*M given by g(v, ·). Let g1 denote the inverse isomorphism. Let j denote the almost complex structure whose action on a given vector v is defined by the rule j v ¼ g1 ðmM ðv; ÞÞ: Note that g(jv, u) ¼ mM(v, u) ¼ g(v, ju). This and the fact that g is symmetric implies that j2 ¼ 1. Here is another way to see this: The almost complex structure j acts on an oriented, orthonormal frame, {e1, e2}, for TM at a given point as je1 ¼ e2 and je2 ¼ e1. It is customary to use T1,0 M to denote the resulting rank 1, complex vector bundle that is defined using this almost complex structure on TM. As explained next, the Levi-Civita covariant derivative on sections of TM induces a covariant derivative on sections of T1,0 M. Here is a quick way to see this: The metric g is covariantly constant, and thus so is g1. Note in this regard that if m is any isomorphism between TM and T*M, then m1 ¼ m1(m) m1. Meanwhile, mM is also covariantly constant. This is proved in Chapter 15.1. As a consequence, j is covariantly constant. This last fact implies that (jv) ¼ jv when v is a section of TM. This confirms that  commutes with j and so defines a covariant derivative for the resulting rank 1, complex bundle T1,0 M. To elaborate on this last point, recall from Chapter 6.3 that T1,0 M sits inside N N N the tensor (TM R C) as the subspace of vectors of the form v 1  jv i. As  commutes with j, it sends a section of this subspace to a section of its tensor product with T*M. Here is another way to see the action of  on sections of E: Fix attention on an open set U  M where there is an orthonormal basis of sections for TM. Let {e1, e2} denote such a basis, but one where mM ¼ e1 ∧ e2. The fact that the N N connection is metric compatible to write e1 ¼ e2 a and e2 ¼ e1 a with N a is a 1-form onN U. The covariant derivative of the dual frame is e ¼ e a 1 2 N and e2 ¼ e1 a. This understood, then (je1) ¼ e2 ¼ e1 a which is j(e1), and likewise, (je2) ¼ j(e2). The curvature 2-form for this covariant derivative for sections of T1,0 M can be written as ik mM where k is a real-valued function on M. The integral over M 1 of  2p k is the integral over M of the first Chern class of the bundle T1,0 M. This was computed in Chapter 14.7 to be 2  2genus(M). The computation in Chapter 14.7 was done using the construction above for a particular choice of metric on TM and thus connection on T1,0M. As the cohomology class of the

16.5

Metrics on manifolds of dimension 2

Chern forms do not depend on the chosen covariant derivative, this value for the integral must hold for any choice of metric. To make contact with the function R, note that T1,0 M is spanned over the N open set U by the section e ¼ p1ffiffi2 ðe1  ie2 Þ; and e ¼ e ia. Thus, the curvature 2-form for the covariant derivative on T1,0 M is given over U by i da ¼ i k e1 ∧ e2. On the other hand, writing da as k e1 ∧ e2 identifies the Riemann curvature tensor component R1212 with k. Thus, R ¼ 2k, and so the integral of R mM is 8p (1  genus(M)).

16.5

Metrics on manifolds of dimension 2

The preceding proposition implies that only the torus can have a Riemannian metric with zero curvature. In the case of a surface with positive genus one can ask if there are metrics where R is constant. As seen previously, there is a metric on S2 with constant R; the metric that is induced from its standard inclusion into R3 as the sphere of radius 1 has R ¼ 2 and the sphere of radius r > 0, the latter has R ¼ 2r2. What follows gives an example of an R ¼ 0 metric on the torus. Let  ¼ {v1, v2} denote an ordered pair of linearly independent vectors in R2. Define a Z2-action on R2 by sending a given pair of integers, (m, n), to 2p(mv1 þ nv2). As explained earlier, the quotient space T ¼ R2/Z2 is diffeomorphic to the standard torus S1  S1 with a flat metric. The standard Euclidean metric on R2 is invariant under translations, so it defines a Riemannian metric on this quotient. The former has R ¼ 0, and thus so does the latter. The next theorem classifies the metrics on compact 2-dimensional manifolds, and in doing so, classifies the metrics with constant scalar curvature. The theorem uses the following terminology: Metrics g and g0 on a given manifold M are said to be conformal if g ¼ f g0 with f a positive function on M. Here, g N and g0 are to be viewed as sections of T*M T*M. More is said about conformal metrics in Chapter 16.6. Theorem 16.5 Let M denote a compact, oriented 2-dimensional manifold, and let g denote a Riemannian metric on M. There exists a positive real function f: M ! R and a diffeomorphism f: M ! M such that  If M ¼ S2, then f f*g is the spherical metric that is obtained by viewing S2 as the unit sphere in R3. The diffeomorphism f is unique up to the action of the group Sl(2; C) on the unit sphere in R3. View S2 as CP1, which is to say the space of complex lines in C2. The group Sl(2; C) acts as a group of C-linear transformations of C2 by definition, and so it acts on the space of complex

229

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16 : The Riemann curvature tensor

lines. The pull-back of the round metric by a diffeomorphism from Sl(2; C) is conformal to the round metric.  If M ¼ S1  S1, then ff*g is a metric with zero curvature, in fact, a metric as described above from the identification S1  S1 ! T for some lattice   R2. The lattice  is unique up to the action of Sl(2; Z) on R2.  If M is a surface of genus greater than 1, then f f*g has constant curvature R ¼ 1. In fact, M is the quotient of the 2-dimensional hyperbolic space (with r ¼ 1) by the action of a discrete subgroup of the Lorentz group. The set of equivalence classes of such hyperbolic metrics can be given a topology such that the complement of a locally finite subspace is a smooth manifold with dimension 6(genus(M)  1). With regards to the third item, the equivalence relation here posits that metrics g and g0 are equivalent if and only if there exists a diffeomorphism f: M ! M such that f*g ¼ g0 . The proof of this theorem requires some tools that this book does not discuss, and so the proof is not given. However, the next part of this chapter says a little bit about changes of the form g ! f g. Meanwhile, Chapter 16.8 discusses the topology of manifolds that admit metrics with constant negative or zero sectional curvature. Both subjects are germaine to the proof of this theorem.

16.6

Conformal changes

Let M be a smooth manifold of any dimension, and suppose that g is a metric on TM. A metric g0 is said to be conformal to g when g0 ¼ f g where f: M ! (0, 1) is a positive function on M. Length measured by g0 differs from those of g, but not angles. Suppose that {ea}1am is an orthonormal frame for T*M as defined by g over an open set U  M. Write h ¼ eu and then (e0 )a ¼ h ea is orthonormal for g0 . This fact can be used to compute the curvature of g0 in terms of derivatives N of h and the curvature of g. To do so write eb ¼ ea Gab with  being the LeviCivita covariant derivative for g. Here Gab ¼ Gba and deb ¼ Gab ∧ ea. (The first condition asserts metric compatibility and the second is the torsion free condition.) I also use here and below the convention that repeated indices are implicitly summed over their range of values. Use 0 to denote the g0 N Levi-Civita covariant derivative and write 0 (e0 )a ¼ (e0 )b (G0 )ba. The equation deb ¼ Gab ∧ ea implies that dðe0 Þb ¼ Gab ∧ ðe0 Þa þ @a hea ∧eb ; where {@ bh} are the components of dh when written in terms of the basis {eb}. Thus, @ bh denotes the directional derivative in the direction of the dual vector

16.6

Conformal changes

field to eb. The left hand side of the last equation must equal (G0 )ab ∧ (e0 )a with (G0 )ab ¼ (G0 )ba. As seen in Chapter 15.2 there is just one way to solve this equation, and here is the solution: ðG0 Þab ¼ Gab þ @b hh1 ea  @a hh1 eb : The curvature 2-form (F0 )ab is given by d(G0 )ab þ (G0 )ac ∧ (G0 )cb, and thus ðF0 Þab ¼ ðF Þab þ ðc ð@b hh1 ÞÞec ∧ea  ðc ð@a hh1 ÞÞec ∧eb þ ð@c hh1 ea  @a hh1 ec Þ∧ð@c hh1 eb  @b hh1 ec Þ: Write this as 12 ðR0 Þabcd ðe0 Þc ∧ ðe0 Þd to obtain the Riemann curvature tensor for g0 : ðR0 Þabcd ¼ e2u ðRabcd þ ðc @b uÞdad  ðd @b uÞdca  ðc @a uÞddb þ ðd @a uÞdcb þ jduj2 ðdac dbd  dad dbc Þ þ @c u@ b udda  @d u@ b udca  @c u@ a uddb þ @d u@ a udcb Þ where the notation is as follows: First, u ¼ ln(h) and dab ¼ 1 if a ¼ b and 0 otherwise. Second, cu ¼ h1@ ch are the components of du when written in terms of the basis {ea}, and a@ cu ¼ @ a@ cu þ Gbac@ bu are the components of du when written in terms of this same basis. Finally, jduj is the norm of the 1-form du as defined by the metric g. The Ricci curvature tensor (R0 )ac can then be written as ðR0 Þac ¼ e2u ðRac  udca  ðn  2Þa @b u þ ðn  2Þðjduj2 dac  a @c uÞÞ; where  is commonly used to denote b@ b. Finally, the scalar curvatures are related by R0 ¼ e2u ðR  2ðn  1Þu þ ðn  2Þðn  1Þjduj2 Þ: If I am looking for a metric g0 conformal to g with constant scalar curvature equal to c; thus R0 ¼ c, then I would have to find a function u on M that solves the equation R ¼ u 

n2 1 jduj2 þ ce2u : 2 2ðn  1Þ

In the case n ¼ 2, this equation reads 1 R ¼ uþ ce2u : 2 As it turns out, it is always possible to solve this equation on a compact, dimension 2 manifold.

231

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16 : The Riemann curvature tensor

A metric, g, on a manifold is said to be locally conformally flat if there exists, in a neighborhood of any given point, a positive function, f, such that f g has vanishing Riemann curvature tensor.

16.7

Sectional curvatures and universal covering spaces

Let M denote a given manifold and g a Riemannian metric for M. The Riemann N curvature is viewed in what follows as a section, Riem, of (∧2 T*M) (∧2 T*M). Fix a point x 2 M, and then a 2-dimensional subspace P  TMjx. Fix a pair of orthonormal vectors u, v in P. The latter define the element u ∧ v 2 ∧2 TMjx. N Evaluate Riem on (u ∧ v) (u ∧ v) to get a number. This number is called the sectional curvature for the subspace P. This number does not depend on the chosen basis vectors u and v for P. What follows is another way to define the sectional curvature. Let {ea} N denote an orthonormal frame for T*Mjx. The section Riem of (∧2 T *M) (∧2 T*M) at x is 1 Riem ¼ Rabcd ðea ∧ eb Þ 4

O

ðec ∧ ed Þ:

Granted this depiction of Riem, the sectional curvature at x for the subspace P is Riemðu; v; u; vÞ ¼ Rabcd ua vb uc vd : Here the notation is such that ua is the pairing of u with ea; this the component of u along the dual vector field to ea. Meanwhile, va is defined analogously using v. By way of an example, if M has dimension 2, then there is just one sectional curvature, and it is equal to 12 R. The sectional curvatures from a metric on a given manifold M can be used in certain instances to say something about its topology. The theorem that follows is a classical result. It refers to the universal cover of a manifold. The latter is defined in Chapter 13.7. Theorem 16.6 Suppose that the metric is such that all sectional curvatures are nonpositive everywhere on M. Then the universal cover of M is diffeomorphic to Rn. What follows sketches a proof of this: Fix a point p 2 M and consider the exponential map expp : TMjp ! M. This map was defined in Chapter 9.2; it says to move a distance 1 along the geodesic that starts at x and has initial velocity vector given by the chosen point in TMjp. The assertion of the theorem follows from the claim that expp is a covering map. The proof of the latter claim has

16.8

The Jacobi field equation

two parts. The first part establishes that the differential of expp is an isomorphism at each point of TMjp. It follows as a consequence of this first part that any given vector v 2 TMjp has a neighborhood that is mapped diffeomorphically by expp onto an open set in X. This last conclusion does not by itself imply that expp is a covering map. Needed is a positive constant, d, with the following property: The inverse image of any given ball in X of radius d is a disjoint union of open sets which are each mapped by expp diffeomorphically to the given ball. The second part of the argument proves that such a constant does indeed exist. Chapter 16.8 gives Part 1 of the argument. Part 2 follows from what is said below in Chapter 16.9.

16.8

The Jacobi field equation

Part 1 of the proof of Theorem 16.6 has five steps. Steps 1– 4 introduce and discuss a novel differential equation called the Jacobi field equation. The final step uses this equation with the assumption of nonnegative sectional curvature to complete Part 1.

Step 1 The image via expp of a given nonzero vector v 2 TMjp is defined using the geodesic through p with tangent vector v at p. This being the case, an analysis of the differential of expp requires first an understanding of how the geodesics through p vary as a function of their tangent vector at p. To investigate this issue, let t ! g(t) denote the geodesic, parameterized by t 2 R with g(0) ¼ p and with g j0 ¼ v 2 TMjp. A deformation of the geodesic g is viewed in what follows as a map f: (1, 1)  R ! M such that f(s, ·) is a geodesic for all s 2 (1, 1), and such that f(0, ·) ¼ g. Let f denote such a deformation. There is no requirement at this point that f(s, 0) ¼ p for all s. Thus, the deformed geodesic is allowed to miss p. Of interest with regards to the differential of expp is the section of g*TM @ at s ¼ 0. This is because the section Z at t ¼ 1 is the image of given by Z ¼ f* @s the vector Zjt¼0 2 TMjp via the differential at v of expp.

Step 2 @ The section Z ¼ ðf* @s Þjs¼0 of g*TM satisfies a certain differential equation by virtue of the fact that each s 2 (1, 1) version of f(s, ·) is a geodesic. The latter equation is called the Jacobi field equation. What follows is a derivation of this

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16 : The Riemann curvature tensor

equation: Fix t* 2 R and a coordinate chart for a neighborhood in X of g(t*). There exists e > 0 such that if jsj < e and jt  t*j < e, then f(s, t) will be in this coordinate chart. For such s and t, use {f1, . . . , fdim(M)} to denote the coordinates of f(s, t) with respect to the given chart. As each f(s, ·) is a geodesic, each will satisfy the differential equation    d2 i d j d k i ¼0 f f f þ G jk dt dt dt2 if jsj < e and jt  t*j < e. Now differentiate this last equation with respect to s and then set s ¼ 0. Doing so finds that the section t ! Z(t) of g*TM obeys an equation that can be written without reference to a coordinate chart as 2t Z þ ðg*F ÞðZ; gÞ ˙ g˙ ¼ 0 where the notation is as follows: First t is the pull-back of the Levi-Civita covariant derivative to the bundle g*TM over R . Second, F is the curvature of the Levi-Civita covariant derivative. Thus, F(u, v) for vectors u and v in TM over a given point is an endomorphism of TM at that point. This last equation is the Jacobi field equation. It is a linear, second-order differential equation for a section of g*TM. A solution is a Jacobi field along g.

Step 3 To see what F looks like, fix an orthonormal, covariantly constant frame t ! {eajt} for g*TM. This is a frame such that t ea ¼ 0 for each index a and each t 2 R, and also such that (g*g)(ea, eb) at any given t 2 R is equal to 1 when a ¼ b and 0 otherwise. Recall from Chapter 11.5 that there exists frames with t ea ¼ 0 for index a and each t 2 R. If the frame at t ¼ 0 is orthonormal, then it will be orthonormal at all times by virtue of the fact that the Levi-Civita connection is metric compatible. Indeed, the latter point implies that d ðg*gÞðea ; eb Þ ¼ ðg*gÞðt ea ; eb Þ þ ðg*gÞðea ; t eb Þ: dt As the right-hand side above is zero if the frame is covariantly constant, so (g*g)(ea, eb) is constant. Write a Jacobi field Z in terms of this frame as Z ¼ Za ea where now Za is a function on R. Here and in what follows, the convention is enforced that repeated indices are summed over their range of values. Meanwhile, g defines a section of g*TM, and so it to can be written as g ¼ g a ea. Since t ea ¼ 0, the Jacobi field equation asserts that the functions {Za} obey the following system of differential equations:

16.8

d2 dt2

Za þ Rabcd g bZc g d ¼ 0;

The Jacobi field equation

ð*Þ

where Rabcd is the function on R which at time t is the indicated component of the Riemann curvature at the point g(t) 2 M when written using the dual frame {eajt}.

Step 4 The next proposition asserts what is needed concerning the question of the existence of Jacobi fields. Proposition 16.7 The set of solutions to the Jacobi equations is a vector space N of dimension twice that of M. This vector space is isomorphic to TMjp TMjp via the map that associates to a solution the vectors (Zjt¼0, tZjt¼0). Proof of Proposition 16.7 This is a consequence of the vector field theorem from Appendix 8.1. To elaborate, introduce the map v: R  (2 TMjp) ! R  (2 TMjp) that sends a given element (t, (u, v)) to (1, v, (Rabcd g b g d)jt uc)). The map z: R ! R  (2 TMjp) that solves this equation with initial value zjt¼0 ¼ (t ¼ 0, u ¼ u,0, v ¼ v0) has components at d time t given by z(t) ¼ (t ¼ t, u ¼ Z, v ¼ dt Z) where Z obeys the Jacobi d field equation with initial value Zjt¼0 ¼ u0 and (dt Z)jt¼0 ¼ v0. Conversely, if Z obeys the Jacobi field equation with these same t ¼ 0 values, then z as just described solves the corresponding vector field equation. The following proposition restates what is said in Step 1. Note in this regard that the geodesic gr : R ! M with grjt¼0 ¼ p and g njt¼0 ¼ r v is given by grjt ¼ gjrt. Proposition 16.8 Let v 2 TMjp denote a given vector with norm 1 and let g: R ! M denote the geodesic with gjt¼0 ¼ p and with g jt¼0 ¼ v. Fix a vector u 2 TMjp and let Z denote the solution to the Jacobi field along g with d Zj0 ¼ 0 and with (dt Z)jt¼0 ¼ u. Suppose that r is a given positive number. Then the differential of expp at r v maps the vector u 2 TMjp to Zjt¼r 2 TMjg(r). Proof of Proposition 16.8 The assertion is a consequence of the definition of the Jacobi field equation.

Step 5 This step indicates why the signs of the sectional curvatures are relevant to the question of whether the differential of expp is or is not everywhere surjective. To start, suppose that g is a geodesic with gjt¼0 ¼ p and suppose that Z is a Jacobi field along g with Z(0) ¼ 0 and Z˙(0) 6¼ 0. Given the second proposition

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in the preceding step, the surjectivity of the differential of expp at points on the ray in TMjp tangent to g jt¼0. follows if it is the case that Zjt 6¼ 0 for all such Z and for all t ¼ 0. To see what to make of this, take the inner product of the expression at time t on the left-hand side of Equation (*) in Step 3 with the vector in Rdim(M) with components {Zajt}1adim(M). The resulting equation can be manipulated so as to read  d 2 1 d2   2 jZj   Z þ 2Rabcd Za g b Zc g d ¼ 0: 2 dt2 dt The term with R is the sectional curvature in the plane spanned by Z and g . If the sectional curvatures are nonpositive everywhere, then this last equation implies that  d 2   2 jZj   Z : 2 dt dt d2

Integrating this equation finds that d jZj2 > 0 dt d since dt jZj is nonzero at t ¼ 0. This shows that jZj2 > 0 for t > 0. Thus, the differential of expp on TMjp on the ray tangent to g b has trivial kernel and so the differential along this ray is at each point an isomorphism. Note in particular that if a 2-dimensional manifold has everywhere negative scalar curvature, then it has negative sectional curvature. As a consequence, its universal cover is diffeomorphic to R2.

16.9

Constant sectional curvature and the Jacobi field equation

A Gaussian coordinate chart centered at a given point in M is defined in Chapter 9.3 using the exponential map. Let p 2 M denote the point in question. To define this chart, fix an orthonormal frame {ea}1an for TMjp, where n here denotes the dimension of M. The inverse of the coordinate chart map sends a point x ¼ (x1, . . . , xn) 2 Rn to the point expp(xaea) 2 M. Here and in what follows, the summation convention for repeated indices is again used. As explained in Chapter 9.3, this map is a diffeomorphism from a ball in Rn about the origin onto a neighborhood of p in M. The last equation in Chapter 9.3 is a schematic formula for the pull-back of the metric via Gaussian coordinates.

16.9

Constant sectional curvature and the Jacobi field equation

A somewhat more explicit formula is derived in the first part of what follows using the Jacobi fields. To start, suppose that g is a geodesic with gjt¼0 ¼ p and with g jt¼0 ¼ v 2 TMjp. As done in the preceding section of this chapter, fix a covariantly constant, orthonormal frame {ea}1an for g*TM so as to write a given Jacobi field along g as Z ¼ Zaea where (Z1, . . . , Zn) is a map from R to Rn that obeys the Equation (*) in that same section of this chapter. The pull-back to T*Mjp of the Riemannian metric by the exponential map is obtained from the solutions to the Jacobi equations as follows: Let u and u0 denote vectors in T*Mjp, and let Z and Z0 denote the corresponding Jacobi fields along g that vanish at t ¼ 0 and are such that tZjt¼0 ¼ u and tZ0 jt¼0 ¼ u0 . Then the inner product at v of u and u0 as defined by expp*g is given by ðZa ðZ0 Þ a Þjt¼1 : To recover what is done at the end of Chapter 9.3, suppose that u·v ¼ uava ¼ 0, d2 then Zava ¼ 0 at all t 2 R since the Jacobi equation implies that dt 2 ðZ vÞ ¼ 0. As a consequence, if v is written as r^ v where r 2 [0, 1) and ^v is a unit vector, then the metric has the form dr

O

dr þ h

where h(u, u0 ) ¼ (Za(Z0 )a)jt¼1. Here, it is assumed that u·v and u0 ·v ¼ 0. Meanwhile Za solves the equation d2 dt2

Za þ r2 Rabcd ^ vb Zc ^ vd ¼ 0

d a Z j0 ua , while Z0 solves the same with initial conditions Za(0) ¼ 0 and dt 0 a d 0 a equation with (Z ) (0) ¼ 0 and dt ðZ Þ j0 ¼ ðu0 Þ a . Note that h(u, u0 ) depends both on r and ^v because this equation depends on them. Suppose now that the sectional curvatures are constants, and in particular are all zero, thus Rabcd ¼ 0. The Jacobi equation asserts in this case that d2 a Z ¼ 0. This equation integrates to give Za ¼ tua. As a consequence, the dt2 pull-back expp*g has the inner product at v of u and u0 equal to u·u0 , which is their Euclidean inner product. This means that the metric on the universal cover of M is the Euclidean metric, and M is obtained from Rn as the quotient by a group of isometries of the Euclidean metric. The following proposition summarizes:

Proposition 16.9 Suppose that M has a Riemannian metric with everywhere zero curvature. Then the universal cover of M is Rdim(M) and M is the quotient of this Euclidean space by a group of isometries of the Euclidean metric.

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Suppose next that Rabcd ¼ c (dacdbd  daddbc) where c is a constant but not zero. The Jacobi equation for Z with u·^ v ¼ 0 reads d2 2

dt

Za þ cr2 Za ¼ 0:

d a Z jt¼0 ¼ ua is In the case where c > 0, the solution with Zjt¼0 ¼ 0 and dt

Za ¼ ðc1=2 rÞ1 sinðc1=2 rtÞua : Thus, the inner product h is given by hðu0 ; uÞ ¼ c1 r2 sin2 ðc1=2 rÞu u0 : This formula for expp*g is identical to the formula for the metric that is obtained using the exponential map on the sphere Sn of radius c1/2. As a consequence, the universal cover of M is the sphere Sn; and M is the quotient of this sphere by a group of isometries of its round metric. In the case when Rabcd ¼ c (dacdbd  daddbc) with c < 0, the solutions are Za ¼ ðjcj1=2 rÞ1 sinhðjcj1=2 rtÞua and the inner product h is given by hðu; u0 Þ ¼ jcj1 r2 sinh2 ðjcj1=2 rÞu u0 : This formula for expp*g is identical to the formula for metric that is obtained using the exponential map on the hyperboloid in R1,n given by t2  x2 ¼ jcj. As a consequence, the universal cover of M is Rn with its hyperbolic metric; this is as described in Chapter 8.4. What follows summarizes the c > 0 and c < 0 cases. Proposition 16.10 Suppose that M has a Riemannian metric with sectional curvatures given by Rabcd ¼ c (dacdbd  daddbc) with c > 0. Then the universal cover of M is Sdim(M) and M is the quotient of the sphere of radius c1/2 by a group of isometries of its round metric. Suppose, on the other hand, that M has a Riemannian metric with sectional curvatures given by Rabcd ¼ c (dacdbd  daddbc) with c < 0. Then the universal cover of M is Rdim(M) and M is the quotient of the Rn by the group of isometries of its hyperbolic metric.

16.10

Manifolds of dimension 3

William Thurston made a profound conjecture about 3-dimensional manifolds that implied a complete list of homeomorphism and diffeomorphism

16.11

The Riemannian curvature of a compact matrix group

equivalence classes. (These are the same in dimension 3.) The conjecture was known as the Geometrization conjecture; and it was proved a few years ago by Grigory Perelman using and extending technology that was developed by Richard Hamilton. It is now known as the Geometrization theorem. What follows is meant to give an idea of what this theorem says. Let M denote a compact, 3-dimensional manifold. Then M can be cut along a finite set of disjoint, embedded 2-dimensional tori and 2-dimensional spheres so that the result is a disjoint union of components. This is to say that these surfaces are deleted from M. The cutting surfaces are determined by the fundamental group of M, and by p2(M), the set of homotopy classes of maps from S2 into M. In particular, the cutting surfaces are defined by algebraic data. This decomposition is known as the JSJ decomposition, named after William Jaco, Peter Shalen and Klaus Johannson. The result of all this cutting has some finite union of components. The Geometrization theorem asserts that each component is the quotient of one, and only one, of eight possible complete Riemannian manifolds by a discrete subgroup of the diffeomorphisms that acts isometrically. Here, a Riemannian manifold is taken to be a pair (M, g) where M is the manifold and g the metric. A diffeomorphism, f: M ! M is said to be an isometry if f*g ¼ g. All but one of these eight Riemannian manifolds can be viewed as a 3-dimensional Lie group with a left-invariant metric. In any event, the list of the eight Riemannian manifolds includes R3 with its flat metric, S3 with its round metric, and the 3-dimensional hyperbolic space with its hyperbolic metric. The list also includes the product of R and S2 with its round metric; and the product of R and 2-dimensional hyperbolic space. There are three more manifolds: The first is the universal cover of Sl(2; R); the second other is the Heisenberg group, this the group of upper triangular 3  3 matrices with all diagonal entries equal to 1. The last is what is called solve geometry. The manifold is R3, but the group law is such that ðx; y; zÞ ðx0 ;y0 ;z0 Þ ¼ ðez x0 þ x; ez y0 þ y; z þ z0 Þ: A very detailed discussion of these geometries can be found in Chapters 3.8 and 4.7 of William Thurston’s book Three Dimensional Geometry and Topology (1997).

16.11

The Riemannian curvature of a compact matrix group

Chapter 8.7 defines a matrix group to be a subgroup of some n  1 version of SO(n) or U(n). Let G denote such connected, matrix group. It is given the

239

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Riemannian metric that is induced from that on SO(n) or U(n) described in Chapter 8.5 or 8.6 as the case may be. This metric is such that there is an orthonormal basis of left invariant vector fields. By way of a reminder, recall from Chapter 5.4 that the tangent space to G at the identity element i is a subvector space of the vector space of either anti-symmetric or anti-Hermitian n  n matrices. The former occurs when G is in SO(n) and the latter when G sits in U(n). This vector space is denoted by lie(G). A given element q 2 lie(G) defines the left-invariant vector field vq that is given at a matrix m 2 G by the formula   T @ : va j ¼ tr q m @m The inner product of vq and vq0 is tr(qTq0 ) when G  SO(n) and tr(q{q) when G  U(n). This inner product defines a Riemannian metric on G. What follows describes the Riemann and Ricci tensors for this metric. For the present purposes, it proves convenient to view a subgroup of U(n) as a subgroup of the larger group SO(2n) so as to give a “one size fits all” account. By way of explanation, recall what is said in Chapter 2.5: The group U(n) is a subgroup of Gl(n; C). Meanwhile, the group Gl(n; C) can viewed as a subgroup of Gl(2n; R). The depiction of Gl(n; C) in Gl(2n; R) makes U(n) and its subgroups appear as subgroups in SO(2n). Use m to denote the dimension of G. Fix an orthonormal basis for lie(G) and let {v1, . . . , vm} denote the corresponding orthonormal basis of left-invariant vector fields. Let {o1, . . . , om} denote the dual basis of left invariant 1-forms. This is to say that ok(vj) is zero if j 6¼ k and one if j ¼ k. The basis {o1, . . . , om} is orthonormal. By way of a reminder from Chapter 5.4 that the left-invariant 1–forms are also in 11 correspondence with the elements in lie(G) with the correspondence such that a given element q 2 lie(G) corresponds to the left-invariant 1-form that is given at m by the rule trðqm1 dmÞ: Keep this formula in mind for what follows. With the basis {o1, . . . , om} fixed, any given covariant derivative on the space of sections of TG corresponds to a set, {Gij}ij,jm, of 1-forms, and vice versa. The correspondence is such that oi ¼ ∑1jm oj

O

aij :

As explained in Chapters 15.1 and 15.3, the covariant derivative  is metric compatible if and only if {aij}1i,jd is such that aij ¼ aji for all index pairs (i, j). Chapter 15.2 introduces the notion of a torsion free covariant derivative. It follows from what is said in Chapter 15.3 that the Levi-Civita covariant

16.11

The Riemannian curvature of a compact matrix group

derivative is determined completely by the preceding antisymmetry conditions {aij ¼ aji}1i,jm together with the m additional conditions doi ¼ ∑1jm aij ∧oj : Write each 1-form aij as a linear combination of the basis elements {o1, . . . , om}, P thus as 1km akij ok where each akij is a function on G. This writes the preceding equation as ij

doi ¼ ∑1j;km ak ok ∧ oj :

ð*Þ

The fact that the basis {oi}1im is invariant under action of left translation implies that the functions {aijk}1i,j,km are constant functions on G. This is so because both the exterior derivative operation that appears on the left side of (*) and the exterior product operation that appears on the right-hand side commute with the pull-back via a smooth map. The fact that {aijk}1i,j,km are constant tremendously simplifies calculations because one need only make computations at the identity element. This is done momentarily. As explained next, the fact that {aijk}1i,j,km are constant also simplifies the task of computing the Riemann curvature tensor. To elaborate, recall from Chapter 15.7 that the 2-form {Fij}1i,jm that determines the curvature 2-form are determined by the 1-forms {Gij}1i,jm by the equation ðF Þij ¼ daij þ S1km aik ∧akj : P Given the defining equation aij ¼ 1kmaijkok and the fact that the functions {aijk}1i,j,km are constant, this last equation finds (F)ij equal to ik kj p q ðF Þij ¼ ∑1k;p;qm ðaijk akq p þ ap aq Þo ∧ o :

The components of the Riemann curvature tensor are determined by the P 2-forms Fij by the rule (F)ij ¼1/2 1p,qm Rijpq op ∧ oq. This being the case, it follows from this last formula that the Riemann tensor, Ricci tensor and scalar curvature function are P ij kp ik kj ik kj  Rijpq ¼ 1km (aijk (akq p þ a k a q þ a p a q  a q a p ). P iq pk ik kq  Ricip ¼ 1k,qm (a k aq þ ap aq ). P kq kp kq  R ¼  1kq,pm (apq k a p þ a p a q ). Note, by the way, that the various functions here are all constant functions on G. It is time now to derive a formula for the constants in the set {aijk}1i,j,km. It proves convenient for this purpose to use the exponential map to parametrize a neighborhood of the identity in G by a ball about the origin in lie(G). Let {q1, . . . qm} denote the given orthonormal basis for lie(G). Given P x ¼ (x1, . . . , xm) 2 Rm, use x·q 2 lie(G) to denote the matrix 1im xi qi. The exponential map associates to any given point x ¼ (x1, . . . , xm) 2 Rm the point

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1 mðxÞ ¼ ex q ¼ 1 þ x q þ ðx qÞ2 þ 2 in G. As the 1-form oi ¼ tr(qim1dm), it follows from this last formula that the pull-back of oi to Rm using the exponential map is the 1-form 1 oi ¼ dxi  ∑1j;km trðqi ½qj ; qk Þxj dxk þ 2 where the unwritten terms involve product of 2 or more of the coordinate functions. Here, the terminology is such that [q, q0 ] denotes the commutator q q0  q0 q. This last formula tells us that oi and doi at the identity element i 2 G are given in these coordinates by the respective forms 1 oi ji ¼ dxi and doi ji ¼  ∑1j;km trðqi ½qj ; qk Þdxj ∧ dxk : 2 These formulas with (*) imply that the functions {a kij}1i,j,km are given by ij

ak ¼

1 trðqi ½qj ; qk Þ: 2

Note, by the way, that the expression on the right here changes sign when any two indices are interchanged. This is so because [q, q0 ] ¼ [q0 , q] for any two matrices, and meanwhile, tr(q [q0 , q00 ]) ¼ tr(q00 [q, q0 ]) for any three matrices. This antisymmetry property allows the components of the Riemann and Ricci tensors, and the scalar curvature function to be rewritten as P ik jk ik jk  Rijpq ¼ 1km (2akijapq k  a p a q þ a q a p ), P iq pq  Ricip ¼ 1k,qm ak ak, P pq  R ¼  1kq,pm apq k ak . As can be seen from the preceding, the scalar curvature is a nonnegative constant which is zero only if all the curvature is zero. Since G is compact, it follows from what is said in Chapter 16.8 that G is the quotient of Rn by a group of isometries. In general, the formula above tells us that the Ricci tensor is nonnegative definite.

16.11.1

The case G ¼ SO(n)

The tangent space is the full vector space of n  n antisymmetric matrices. An orthonormal basis is fp1ffiffi2 qab g1a

T*M ¼

>

>

p

T0;1 MÞ:

N

The summand (∧m T1,0M) (∧q T0,1M) is denoted by T m,qM. These are said to be forms of type (p, q). Note that the ∧2n T*MC ¼ Tn,nM, this a complex, rank 1 line bundle that is isomorphic to M  C. The standard example for an almost complex structure comes from R2n and its canonical almost comple structure, j0, that acts so that j0 1,0

@ @x2a1

¼

@ : @x2a

In this case, T M is spanned by {dzk ¼ dx2k1 þ i dx2k}1kn. Other examples come from complex manifolds.

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17 : Complex manifolds

17.1

Some basics concerning holomorphic functions on Cn

The definition of a complex manifold requires the brief digression that follows to introduce some definitions from complex function theory. To start the digression, fix an integer n  1 and use j0 to view R2n as Cn. Keep in mind that the complex coordinate functions, (z1, . . . , zn), on the Cn are defined in terms of the real coordinates (x1, . . . , x2n) on R2n by the rule zk ¼ x2k1 þ ix2k. A smooth map, f ¼ (f1, f2), from a domain in R2n to R2 ! R2 can be viewed as a complex valued function on a domain in Cn when R2n is viewed as Cn and R2 is viewed as C. Such a function is said to be holomorphic when @@zk f ¼ 0 @ for each k 2 {1, . . . , n}. Here, @ @z k ¼ 12 ð@x2k1 þ i @x@2k Þ. This is to say that the R2 components (f1, f2) of f obey the Cauchy–Riemann equations @f 1 @f 2  ¼0 @x2k1 @x2k

@f 2 @f 1 þ ¼0 @x2k1 @x2k

and

for each k 2 {1, . . . , n}. For example, a function f that is given as a polynomial or convergent power series in the complex coordinates (z1, . . . , zn) is holomorphic. Note that the Cauchy–Riemann equations say the following: View f again as a map from R2n to R2. View the differential of f at each point of its domain as a linear transformation, f : R2n ! R2, thus as a matrix with 2 rows * and 2n columns. Then f j0 ¼ j0 f : *

*

This is to say that f intertwines the action of the 2n-dimensional version of * j0 with that of the 2-dimensional version. Indeed, this follows by virtue of the fact that columns 2k  1 and 2k of the differential f are *   @2k1 f 1 @2k f 1 ; @2k1 f 2 @2k f 2 where @ i is shorthand here (and in what follows) for @x@ i . In particular, the latter has the form   ak bk ¼ ak i þ bk j0 bk ak @f if and only if f is holomorphic. Note, by the way that ak þ ibk here is @z where k @ 1 @ @ ¼ ð  i Þ: @zk 2 @x2k1 @x2k

If m  1 is another given integer, then a map f from a domain in R2n to R2m can be viewed as a map from a domain in Cn to Cm. Such a map is said to be holomorphic if its entries, ( f1, . . . , fm) 2 Cm ¼ m C, are each holomorphic functions on its domain in Cn. Here is an equivalent definition: View f in its

17.2

The definition of a complex manifold

original guise as a map from R2n to R2m. Its differential at each point is a matrix, f , with 2m rows and 2n columns; and as in the m ¼ 1 case, f is * holomorphic if and only if f*j0 ¼ j0 f*. To see this, note that the j’th row of the matrix f has k’th entry @f j =@xk and so f* can be written in blocks of size * 2  2 with the block corresponding to rows 2j  1 and 2j, columns 2k  1 and 2k given by     aj;k bj;k @2k1 f 2j1 @2k f 2j1 ¼ bj;k aj;k @2k1 f 2j @2k f 2j where the equality holds if and only if f is holomorphic. Note that the condition f j0 ¼ j0 f means that f can be viewed as a C-linear transformation from * * * Cn to Cm; which is to say a matrix with m columns and n rows whose entries are complex numbers. This matrix is denoted by @ f. The C-valued entry for the j’th @f row and k’th column of @f is @zkj ¼ aj;k þ ibj;k with the numbers aj,k and bj,k as above. In the particular case where m ¼ n, the condition f j0 ¼ j0 f means that * * @f at each point defines an element in M(n; C), this the space of n  n matrices with complex number entries. As a parenthetical remark, note that the matrix @ f in M(n; C) is invertible if and only if the 2n  2n real matrix f* is invertible. This follows from the fact that det(f ) is equal to jdet(@ f )j2 where det(@ f ) signifies the determinant of the * C-valued matrix @ f . This means that when checking the invertibility of a holomorphic map from Cn to Cn, one can work with the holomorphic coordinates instead of the real coordinates.

17.2

The definition of a complex manifold

Turn now to the definition of a complex manifold. To start, fix an integer n  1 and let M denote a smooth manifold of dimension 2n. A complex manifold structure on M is defined by a coordinate cover U with a certain special property. To elaborate, keep in mind that any given element from U is a pair (U, jU) where U  M is an open set and jU : U ! R2n is a diffeomorphism. These pairs from U must be such that for each (U, jU) 2 U, there is a diffeomorphism, cU, from R2n onto an open set in R2n such that the following condition is satisfied: Let (U, jU) and (V, jV) denote any two pairs from U. Use j0 to identify R2n with Cn. Let fU ¼ cU  jU and likewise define fV . View the transition function n jV  f1 U as a map from one domain in C another. This map must define a holomorphic function on its domain.

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At the risk of being overly pedantic, this means the following: For each k 2 {1, . . . , n}, define the complex coordinate zk from the real coordinates (x1, . . . , x2n) by zk ¼ x2k1 þ ix2k. To say that fV  f1 U is holomorphic means that @ @z k ðfv  f1 Þ must equal zero at each point of the domain of fV  f1 U and U for each k 2 {1, . . . , n}. With what is said in Chapter 17.1 in mind, what follows is another way to present the constraint on U: Let (U, jU) and (V, jV) denote any two pairs from U. Then the differential of 1 must commute with j0 at each point of the domain the corresponding fV  fU of fV  f1 U . Two complex structures on M are said to be equivalent when there is a diffeomorphism, c: M ! M with the following property: Let (U, jU) denote a chart for the first complex structure, and let (U0 , jU0 ) denote a chart for the second if c(U) intersects U0 , then fU0  c  f1 U defines a holomorphic map between one domain in Cn and another. It is customary to view a complex manifold as a manifold with an equivalence class of complex structures. To make contact with what was said at the end of this chapter’s introduction, keep in mind that the differential of the coordinate transition functions give the vector bundle transition functions for TM. The fact that these commute with j0 implies that TM has the structure of a complex, rank n vector bundle. Said differently, the tangent bundle of a complex manifold has a canonical almost complex structure, this given over the open set from an element of U by j ¼ (fU1) j0(fU) . Here, (fU) : TMjU ! TRn denotes the differential of * * * the coordinate chart fU.

17.3

First examples of complex manifolds

Of course, Cn, and any open set in Cn is a complex manifold. There are others. Example 17.1 Introduce from Chapter 6.8 the space CPn of complex, 1-dimensional subspaces in Cnþ1. As noted in Chapter 10.5, it is also the quotient space S2nþ1/S1. It is also the quotient space (Cnþ1{0})/C* where C* ¼ C{0} is the multiplicative group of nonzero complex numbers. This group acts on Cnþ1 by the rule (u, z ¼ (z1, . . . , zn)) ! uz ¼ (uz1, . . . , uzn). This view of CPn is equivalent to the other because C* ¼ Rþ  S1 where Rþ ¼ (0, 1) and (Cnþ1{0})/Rþ ¼ S2nþ1. As noted in Example 10.4, the manifold CP1 is also S2. In general, the manifold CPn is an example of a complex manifold. To see this complex structure, recall from Chapter 6.8 that it can be covered by nþ 1 coordinate charts. The k’th chart consists of the 1-dimensional subspaces of the 6 0. Let O k denote this form C z  Cnþ1 where z ¼ (z1, . . . , znþ1) is such that zk ¼

17.3

First examples of complex manifolds

chart. As explained in Chapter 6.8, a diffeomorphism jk: O k ! Cn sends z to the n-tuple whose j’th component for j < k is zj / zk, and whose j’th component 1 for j  k is zjþ1/zk. The transition function jk  jm is defined as follows: Assume that m < k. The transition function sends (u1, . . . , uk) to the point whose j’th component is uj / uk1 for j < m, whose m’th component is 1/uk1, any given j 2 {mþ1, . . . , k1} component is uj1/uk1, and any j  k component is uj/uk1. This transition function is holomorphic on its domain of definition. Note that in this case, the U ¼ O k version of the map cU can be taken to be the identity map. Example 17.2 Fix integers m > n  1 and introduce from Chapter 6.8 the complex Grassmannian GrC(m; n) whose points are the n-dimensional complex subspaces in Cm. It follows from the description given in Chapter 6.8 that this Grassmannian is a complex manifold. To elaborate, Chapter 6.8 gives a coordinate chart cover of GrC(m; n) with the coordinate chart map sending a point in the coordinate chart to the version of Cn(mn) that is obtained by taking the entries of a complex n(m  n) matrix. Moreover, the transition functions given in Chapter 6.8 between any two such charts are holomorphic functions of the entries of the matrix. This implies that GrC(m; n) is a complex manifold. Here, again, the relevant version of the map cU is the identity map. Example 17.3 An even dimensional torus, T2n ¼ 2n S1, has the structure of a complex manifold. This can be seen as follows: One can view S1 as the quotient space R / (2pZ) with the quotient map sending any given t 2 R to the unit length complex number eit. The product of such maps identifies 2n S1 with R2n/(2pZ2n). View R2n in the standard way as Cn. This done, the quotient map identifies points z and z0 from Cn if and only if z0 ¼ z þ 2p n where n 2 Cn is a vector whose real and imaginary parts have integer entries. It follows from this that the torus T2n has a coordinate atlas U such that each (U, jU) 2 U is such that U is the image via the projection from Cn of a ball in Cn centered with radius less than 2p. In this case, the map fU can be taken to be the identity map on this ball. This being the case, then the transition function fV  f1 U has the form z ! z þ 2p n where n 2 Cn is a fixed vector whose real and imaginary parts have integer entries. I hope it is evident that such a translation is a holomorphic map. Example 17.4 A surface of genus greater than one has the structure of a complex manifold. The genus zero case is S2 which is CP1 and so has a complex structure. The genus one case is the torus T2 and so a surface of genus one also has a complex structure. What follows describes how to construct a complex manifold structure for a surface of genus g > 1. The construction does not speak of the surface as embedded in R3 or, for that matter, in any other ambient manifold. To set the stage however, picture the following construction of a surface of genus g in R3: Start with a round ball of modeling clay, so the

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boundary is S2. Now, make g solid, reasonably long cylinders of clay, bend each one into a U shape, and then attach it to the ball by sticking its ends to the boundary of the ball using 2g separate disks on the boundary for the attaching regions—one disk for each end of each cylinder. Smooth the creases where the attachments are made, and—Voila`—the boundary of what you have is a surface of genus g. I am going to make this same handle attaching construction without reference to R3 and without reference to the interiors of either the balls or the solid cylinders. To elaborate, I describe in what follows a surface of genus g as a union of g þ 1 open sets, {U0, U1, . . . , Ug}. The set U0 will be diffeomorphic to the complement in S2 of 2g disjoint, small radius disks. Each of the other sets is diffeomorphic to the annulus in C where the complex coordinate z obeys e2 < jzj < e2, where e here denotes Euler’s constant, whose natural log is 1. To put this in perspective with regards to the modeling clay construction, the set U0 will represent the boundary of the original ball of clay that remains visible after the solid clay cylinders are attached. To identify the role of the annuli, I point out that a solid cylinder of the sort I described in the modeling clay discussion can be viewed (before being bent into a U shape) as the subset of R3 where the Euclidean coordinates (x1, x2, x3) obeys x21 þ x22  1 and 1  x3  1: The boundary of this solid cylinder can be decomposed into three parts. The first two are the disks where x3 ¼ 1 and x3 ¼ 1. These are the regions that are glued to the boundary of S2 in the modeling clay construction. The remainder of the boundary is the set of points where x21 þ x22 ¼ 1 and 1  x3  1. The part of this boundary where 1 < x3 < 1 is a smooth manifold that is diffeomorphic to the annulus in C where e1 < jzj < e; the diffeomorphism sends z 2 C to ðx1 ¼ cosðz=jzjÞ; x2 ¼ sinðz=jzjÞ; x3 ¼ lnjzjÞ: I have described gþ 1 open sets {U0, U1, . . . , Ug}. I describe next the identifications that are used to construct a surface. To begin, I must say more about U0. View S2 as CP1, and as such it has an open cover with two sets, both diffeomorphic to C. The first set I denote as O 0. It consists of the complex lines through 0 in C2 that are the linear span of a vector that can be written as (1, u) with u 2 C. The value of u gives the coordinate in C for this open set. The other open set consists of the linear span of the vectors through the origin in C2 that can be written as (w, 1) where w 2 C gives the complex coordinate on this open set. The transition function identifies w ¼ 1/u where both u and w are nonzero. Fix 2g points in C that are pair-wise separated by distance at least e10. The set U0 is the complement in the CP1 of the subset of points in the coordinate chart O 0 where the coordinate u has distance e2 or less from any

17.4

The Newlander–Nirenberg theorem

one of these 2g points. Group these 2g points into g sets of two points, {(pk, qk)}1kg. Now consider the identifications between the various open sets from the collection {U0, U1, . . . , Ug}. The identification will pair points in any given k > 0 version of Uk with points in U0. If k 6¼ k0 are both greater than zero, no point in Uk will be paired with a point in Uk0 . This understood, I will describe only the identifications that concern the set U1 as the description for the other sets is the same but for the notation. I use (p, q) in what follows to denote the pair (p1, q1). The identification has two steps, one for near p and the other for near q. Step 1: The part of the disk of radius e1 centered at p that lies in U0 is the annulus centered at p where e2 < ju  pj < e1. This annulus is identified with the part of the cylinder U1 where e2 < jzj < e1 by the rule z ¼ e3 =ðu  pÞ: Step 2: The part of the disk of radius e1 at q that lies in U0 is the annulus centered at p where e2 < ju  qj < e1. This annulus is identified with the part of the cylinder U1 where e < jzj < e2 by the rule z ¼ 1=ðu  qÞ: Let  denote the space obtained by making all of these equivalences. It follows from what is said in Chapter 1.5 that the  has the structure of a smooth manifold, one where the quotient map from each set {U0, U1, . . . , Ug} to  defines an embedding onto an open set in . This understood, I now view each of these sets as sitting in . The complex manifold structure for  comes from that on each of these sets. Indeed, I presented each as an open subset of a complex manifold, U0 in CP1 and each k > 0 version of Uk in C. The transition functions between these open sets are relevant only for Uk U0, and the latter are the identification maps in Steps 1 and 2 above. In particular, they are holomorphic maps on their domains of definition. I leave it to you to verify that  is a surface of genus g. (You might try mimicking the construction in Example 14.9 to construct a connection on the complex line bundle T1,0, and then compute the resulting first Chern class.) ¨

17.4

The Newlander–Nirenberg theorem

Suppose that M is a 2n-dimensional manifold and that j is a given almost complex structure on TM. The question arises as to whether j comes from an underlying complex structure on M. The Newlander–Nirenberg theorem gives necessary and sufficient conditions for this to occur.

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To set the stage for this theorem, consider that Chapter 12.1’s exterior derivative, d, acts on sections of T1,0M since such a section is a C-valued 1-form; albeit one with some special properties with regards to j. In any event, if n is a section of T1,0M, then da will be a C-valued 2-form on M which is to say a section of ∧2 T*MC. As such, it decomposes into forms of type (2, 0), (1, 1) and (0, 2). Introduce Nj n ¼ ðdnÞð0;2Þ to denote the (0, 2) part of dn. Note that Nj(f n) ¼ f Nj n for any function f. This is because d(f n) ¼ df ∧ n þ f dn. In particular, as n is of type (1, 0), so df ∧ n has no components of type (0, 2). The fact that Nj(fn) ¼ f Njn for all functions f implies that Nj defines a section of Hom(T 1,0M; T 0,2M). This follows from the Lemma 11.1 in Chapter 11.2 as applied to the case where E ¼ T1,0M, E0 ¼ T 0,2M and L¼ Nj. This section Nj of Hom(T1,0M; T 0,2M) is called the Nijenhuis tensor. Note that Nj ¼ 0 if j comes from a complex manifold structure on M. Here is how to see this: Let  denote a set of charts that define the complex manifold structure, let U   and let jU denote the map to R2n. Since Nj is a tensor, it is enough to check that it acts as zero on a local basis for T1,0M. Such a local basis is {ak ¼ jU*dzk}1kn. Then dak ¼ 0 so (dak)0,2 ¼ 0. Theorem 17.5 (Newlander and Nirenberg) An almost complex structure on M comes from a complex structure if and only if its Nijenhuis tensor is zero. To state an equivalent condition, remember that a section of T1,0M is a C-valued vector field on M on which j acts as multiplication by i. The commutator of two vector fields is also a vector field, and so one can ask if the commutator of two sections of T1,0M is a section of T1,0M or not. Does the commutator have any T0,1M part? With the preceding understood, here is the equivalent condition: The Nijenhuis tensor Nj ¼ 0 if and only if the commutator of two sections of T1,0M is also a section of T1,0M. To prove this, use the relation between commutator and exterior derivative given by the formula (da)(v, v0 ) ¼ a([v, v0 ]) þ va(v0 )  v0 a(v). Here, a(v) is viewed as the pairing between a 1-form and a vector field, thus a function on M. Meanwhile, v0 a(v) denotes the directional derivative of the function a(v) in the direction of v0 . The proof of the Newlander–Nirenberg theorem would take us some way into complex analysis, and so I won’t say more about it. See A. Newlander and L. Nirenberg, “Complex analytic coordinates in almost complex manifolds,” Annals of Mathematics, Second Series 65 (1957), 391–404. What follows is an immediate corollary: Theorem 17.6 Let M denote an oriented manifold of dimension 2. Then every almost complex structure on TM comes from a complex manifold structure.

17.4

The Newlander–Nirenberg theorem

The reason is that ∧2 TMC ¼ ∧1,1TMC in dimension 2. There are neither (2, 0) or (0, 2) forms. By the way, every oriented, 2-dimensional manifold has a plethora of almost complex structures for its tangent bundle as each Riemannian metric determines one. This almost complex structure can be seen as follows: Take an oriented, orthonormal frame for TM near any given point. Let {e1, e2} denote this frame. Then je1 ¼ e2 and je2 ¼ e1. This doesn’t depend on the choice of frame because j as just defined commutes with any the action of a matrix in SO(2) on the set of oriented, orthonormal frames. Indeed, a new frame {e10 , e20 } is related to the old by e10 ¼ cosy e1 þ siny e2 and e20 ¼ siny e1 þ cosy e2. Hence, je10 ¼ cosy e2  siny e1 ¼ e20 . Note as well that j depends only on the conformal class of the given metric. To see this, suppose that h > 0 and g0 ¼ h2g. Then if {e1, e2} are orthonormal for g, then the pair {he1, he2} are orthonormal for g0 . This understood, the g and g0 versions of the almost complex structure both map e1 to e2 and e2 to e1. To indicate what a proof of the Newlander–Nirenberg theorem involves in dimension 2, consider the following situation: Suppose that j is a given almost complex structure on a surface. Fix a point p and a coordinate chart centered at p so as to identify a neighborhood of p with a disk centered at the origin in R2. View R2 as C with the usual complex coordinate z ¼ x þ iy coordinate on C. Let dz ¼ dx þ idy and dz¯ ¼ dx  idy denote the standard basis for the respective j0 versions of T1,0C and T 0,1C. The given almost complex structure, j, appears in this coordinate chart as an almost complex structure on the disk in C. The latter’s version of T1,0C will be spanned by a complex number valued 1-form that can be written in terms of the basis dz and dz¯ as e ¼ adz þ bd z where a and b are C-valued functions that are constrained so that jaj and jbj are nowhere equal. Here is why they can’t be equal: The C-valued 1-form e and its complex conjugate e¯ are supposed to be linearly independent over C because they must span at each point the fiber of the 2-complex dimensional vector bundle TCjC. However, these to C-valued 1-forms are linearly independent over C at a given point if and only if jaj 6¼ jbj at this point. Note in this regard that e ¼ (a/b¯) e¯ if e and e¯ are linearly dependent; and this equation can hold if and only if jaj ¼ jbj. With the preceding understood, suppose that jaj > jbj. The discussion in the alternative case is the same once z and z¯ are switched. If j comes from an underlying complex structure, then there is a coordinate chart map that gives a diffeomorphism from a neighborhood of the origin in C to R2 sending 0 to 0 and with the following property: Write the map as z ! w(z) ¼ w1(z) þ i w2(z), thus as a C-valued function. Use dw to denote the C-valued 1-form dw1 þ idw2. Then j’s version of T1,0C is spanned by dw. This is to say that e can be written

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as a dw with a a nowhere zero, C-valued function. To see what this means for w, use the Chain rule to write dw ¼

@w @w dz þ d z : @z @ z

Note in this regard that this version of the Chain rule amounts to no more than using the standard 2-variable, real version to first write dw1 þ idw2 as     @ @ @ @ w1 þ i w2 dx þ w1 þ i w2 dy; @x @x @y @y @ @ then write dx as 12 ðdz þ d z Þ and dy as 12 ðd z dzÞ; and finally write @x as @z þ @@z @ @ @ and @y as ð@z  @ z Þ. In any event, with dw written as above, and with e ¼ adz þ bdz¯, you can see by directly comparing respective dz and dz¯ components that e can be written as a dw if and only if the C-valued function w obeys the equation @w a @w ¼ : @ z b @z

This said, the proof of the Newlander–Nirenberg theorem in this context amounts to proving that there does indeed exist a diffeomorphism from a neighborhood of the origin to a neighborhood of the origin in C that satisfies the preceding equation. (This existence theorem was known in the 2-dimensional case before the work of Newlander and Nirenberg: It can be proved using a version of the contraction mapping theorem (Theorem 1.6) with the help of some basic facts about the Green’s function for the Laplacian on R2.) In dimension 4, there are manifolds that admit almost complex structures but no complex structures. For example, the connect sum of 3 copies of CP2. (The connect sum of two copies of CP2 has no almost complex structures: There is a topological obstruction to the existence of a square 1 endomorphism of its tangent bundle.) A famous open question concerns S6: This manifold has almost complex structures, but it is not known whether S6 can be given the structure of a complex manifold. What follows describes an almost complex structure for TS6 that comes from the octonion algebra. The vector space, O, of octonions is 8-dimensional; it can be viewed as the space of pairs (a, b) where a and b are quaternions. Multiplication is defined by the rule: ða; bÞ ðc; dÞ ¼ ðac  d{ b; da þ bc{ Þ: Here, the notation is as follows: Write a quaterion q as q0 þ qiti where (q0, q1, q2, q3) are real and t1, t2 and t3 are the Pauli matrices from Chapter 6.4. Then q{ ¼ q0  qiti. Thus, qq{ ¼ jqj2. This algebra is nonassociative.

17.6

The almost Ka¨hler 2-form

Terminology is such that q0 is called the real part of q and qjtj is the imaginary part. Identify R7 as the subspace of pairs (a, b) 2 O where a has zero real part. Identify S6 as the unit sphere in S7, thus the set of pairs (a, b) 2 O with real(a) ¼ 0 and jaj2 þ jbj2 ¼ 1. This done, then the tangent space to S6 at a given point (a, b) consists of the vector subspace in O whose elements are pairs (c, d) where both c and (a{c þ b{d) have vanishing real part. Granted this identification of TS6j(a,b), a linear map j(a,b): TS6j(a,b) ! O is defined by the rule jjða;bÞ ðc; dÞ ¼ ða; bÞ ðc; dÞ ¼ ðac  d{ b; da þ bc{ Þ: I leave it as an exercise to verify that j maps TS6 to itself and has square 1.

17.5

Metrics and almost complex structures on TM

Suppose that M is a 2n-dimensional manifold with an almost complex structure, j, and metric g. If the metric g is such that j acts as an orthogonal transformation, then j and g are said to be compatible. Note that j acts orthogonally if and only if gðu; jvÞ ¼ gðju; vÞ: The metric g defines a Hermitian metric on the complex vector bundle T1,0M if and only if the preceding condition is obeyed. To see why this is, remember that vectors in T1,0M are of the form u ¼ u  ij·u with u a vector in TM. This understood, the Hermitian metric on T1,0M is defined so that the inner product of u and v ¼ v  i jv is   hu; vi ¼ gðu; vÞ þ gðju;jvÞ þ i gðju; vÞ  gðu; jvÞ : The condition for defining a Hermitian form is that the complex conjugate of hu, vi must equal hu, vi. As you can see, this is the case for all u and v if and only if g(u, jv) and g(ju, v) have opposite signs. What follows is a converse to the preceding observation: A Hermitian metric h , i on T1,0M defines a Riemannian metric on TM by setting g(u, v) to equal 12 of the real part of hu, vi. The almost complex structure j defines an orthogonal transformation with respect to this metric.

17.6

The almost Ka¨hler 2-form

Given an almost complex structure on TM, the standard default option is to choose the metric g so that j acts as an orthogonal transformation. Assuming

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that such is the case, then the bilinear form that assigns g(u, jv) to vectors u and v is antisymmetric. Thus, it defines a 2-form on M, which is to say a section of ∧2 T*M. This 2-form is denoted in what follows by o. It has the nice property that its n’th power, on, is n! times the volume form that is defined by the metric. Remember in this regard that this volume form is given at any given point in terms of an orthonormal basis {e1, . . . , e2n} by O ¼ e1 ∧ . . . ∧ e2n. To see why o has this property, choose orthonormal basis for T*M at a given point so that je2k1 ¼ e2k and je2k ¼ e2k1 for each k 2 {1, . . . , n}. Written with respect to this basis finds that o ¼ ∑1kn e2k1 ∧e2k ; and so taking the wedge product of n copies of o finds on ¼ n! e1 ∧ . . . ∧ en. Note that this implies that a manifold with an almost complex structure has a canonical orientation, this as defined by on. The form o is called the almost Ka¨hler form unless j comes from a complex structure on M and o is a closed 2-form. In this case, o is called the Ka¨hler form, and M is called a Ka¨hler manifold. Suppose that j is an almost complex structure on M. If o is a given 2-form on M, then j is said to be o-compatible when the bilinear form o(·, j(·)) on TM defines a Riemannian metric. The complex structure j acts as an orthogonal transformation for this metric, since g(u, jv) ¼ o(u, j2v) ¼ o(u, v). As o is antisymmetric, o(u, v) ¼ o(v, u), which is o(v, j2(u)), and thus g(v, ju). Since g is a metric, it is symmetric, so we see that g(u, jv) ¼ g(ju, v), thus proving that the transpose of j is j. Since j(j) is the identity, j’s transpose is its inverse and so j is orthogonal. It then follows that this metric defines a Hermitian metric on T1,0M and o is its almost Ka¨hler form. Thus, one can go from 2-forms and almost complex structures to metrics, or from metrics and almost complex structures to 2-forms. By the way, if one is given only a 2-form o whose top exterior power is nowhere zero, there is in all cases a complex structure, j, on M such that the bilinear form given by o(·, j(·)) is a Riemannian metric. The proof that o-compatible almost complex structures exist is given in Appendix 17.1.

17.7

Symplectic forms

A closed 2-form, o, on an even dimensional manifold is said to be symplectic if its top exterior power is nowhere zero. There is a cohomology obstruction that must vanish if o is to be closed. To see this, note that the integral of on over

17.8

Ka¨hler manifolds

M is nonzero. This has the following consequence: The class in H2De Rham(M) that is defined by o must be nonzero, as must all of its powers. In particular, if the De Rham cohomology of M has no 2-dimensional class with nonzero n’th power, then there can be no symplectic form on M. For example, there are no closed, almost Ka¨hler forms on S6. There may be other obstructions to the existence of a symplectic form.

17.8

Ka¨hler manifolds

Suppose that M is a 2n-dimensional manifold with an almost complex structure j. As noted above, it is natural to choose a metric, g, such that j acts in an orthogonal transformation. The question then arises as to whether g can be chosen so that j is also covariantly constant with respect to the g’s Levi-Civita connection. These sorts of metrics can be viewed as being maximally compatible. As it turns out, a metric g that is maximally compatible in this sense may not exist: Proposition 17.7 If j and a metric g are compatible and j is covariantly constant with respect to the Levi-Civita covariant derivative, then j’s Nijenhuis tensor is zero. As a consequence, j, comes from a complex manifold structure. Proof of Proposition 17.7 To see why, remark that if j is covariantly constant with respect to a covariantNderivative, , on TM, then  maps sections of T1,0M to sections of T1,0M T*M. To elaborate, suppose that {uk}k¼1,2, . . . ,n is a basis of sections for T1,0M on some open set in M. Then a randomly chosen covariant derivative will act so that uk ¼ um

O

Bmk þ  um

O

Cmk

where each Bmk and Cmk are 1-forms. Here and in what follows, the summation convention is employed whereby repeated indices are implicitly to be summed over their range of values, {1, . . . , n}. The condition that j is covariantly constant means the condition that (juk) ¼ juk. Since juk ¼ iuk, and since juk ¼ jum

O

Bmk þ j um

O

Cmk ¼ ium

O

Bmk  ium

O

Cmk ;

the condition that j commute with  requires that Cmk ¼ 0 for all indices 1  a, b  n. On the other hand, suppose that  is a torsion free connection. This means that dua ¼ dua and so

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duk ¼ Bmk ∧um  Cmk ∧ um: Thus, the Nijenhuis tensor can be obtained from the (0, 1) parts of the 1-forms {Ckm}. Indeed, writing Cmk ¼ cþmki ui þ cmki ui, one sees that the Nijenhuis tensor is given in this basis by cmki  um ∧  ui. In particular, if C ¼ 0, then the Nijenhuis tensor is zero. If j is covariantly constant with respect to the metric’s Levi-Civita covariant, then so is the 2-form o(·, ·) ¼ g(·, j(·)). (Remember that the metric is also covariantly constant.) Since the covariant derivative is torsion free, this implies that o is also closed, and so j and the metric g give M the structure of a Ka¨hler manifold.

17.9

Complex manifolds with closed almost Ka¨hler form

As explained in the previous part of this chapter, if M has an complex structure, j, with compatible metric g such that j is covariantly constant with respect to the Levi-Civita covariant derivative, then the Nijenhuis tensor of j vanishes and so j comes from a complex manifold structure. The previous part of the chapter also explained why the almost Ka¨hler 2-form o is closed in this case. A converse of sorts to this conclusion is also true: Proposition 17.8 Suppose that j comes from a complex manifold structure on M and that g is a compatible metric such that the associated almost Ka¨hler form o is closed. Then j and o are both covariantly constant. Proof of Proposition 17.8 Fix an orthonormal frame {e1, . . . , e2n} for T*M on a neighborhood of a given point, one such that je2k1 ¼ e2k. Write N eb ¼ ea Wab where each Wab is a 1-form, and where Wba ¼ Wab. Here,  denotes the Levi-Civita connection. Introduce the complex basis {uk ¼ e2k1 þ ie2k} for the dual space, T1,0M of T1,0M. A computation finds uk ¼ um

O

Bmk þ  um

O

Cmk

where

    Bmk ¼ 12 Wð2m1Þð2k1Þ þ Wð2mÞð2kÞ þ i Wð2m1Þð2kÞ  Wð2mÞð2k1Þ     Cmk ¼ 12 Wð2m1Þð2k1Þ  Wð2mÞð2kÞ  i Wð2m1Þð2kÞ þ Wð2mÞð2k1Þ : Note that Bkm ¼ Bmk , so the matrix of 1-forms, B, with entries {Bmk} is antiHermitian. Meanwhile, Ckm ¼ Cmk. Write the component of Cmk in T0,1M as cmki u¯ i. The condition that j have zero Nijenhuis tensor requires that cmki ¼ cikm. Given that cmki ¼ ckmi, this requires that cmki ¼ 0 for all

17.10

Examples of Ka¨hler manifolds

indices k, m and i. Essentially the same calculation was done in Chapter 15.3 to prove that there is but a single torsion free, metric compatible connection. Meanwhile, the form o can written as o¼

i k uk : u ∧ 2p

Thus,

  i mk mk m k m k k m k m km km B ∧ u ∧  u C ∧  u ∧ u  B ∧u ∧ u  C u ∧ u : do ¼ 2p

This uses the fact that  is torsion free. The terms involving the matrix B and its conjugate vanish because B is an anti-Hermitian matrix of 1-forms. The term with C cannot cancel the term with its conjugate as the C-term is a section of T1,2M and its conjugate is a section of T2,1M. Given that Ckm ¼ Cmk, the C term is zero if and only if C ¼ 0 identically. These last conclusions tell us that uk ¼ um

O

Bmk

and so the Levi-Civita covariant derivative commutes with j. This is because the preceding equation asserts that if u is a section of T1,0M, then u is a 1-form on M with values in T1,0M. In particular, j is covariantly constant. This means that o too is covariantly constant because the metric g is covariantly constant. One can see this directly because with the C-term absent in the covariant derivative of u, one has   O O O O i mk m k k m km  u o ¼ ; u u B þu B 2p and this is zero because B is an anti-Hermitian matrix of 1-forms.

17.10

Examples of Ka¨hler manifolds

What follows are some examples of Ka¨hler and non-Ka¨hler complex manifolds. Example 17.9 The manifold S1  S3 is a nonexample. It is an example of a manifold with a complex structure but no Ka¨hler structure. (There is no 2-dimensional cohomology.) To see the complex structure, note that C2{0} is diffeomorphic to R  S3 with the diffeomorphism given as follows: Identify S3 with the unit sphere in C2. Then the diffeomorphism sends a pair (t, (a, b)) 2 R  S3 to (et a, et b) 2 C2{0}. View the manifold S1  S3 as (R /Z)  S3

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where the Z-action on R has the integer n acting to send a given t 2 R to t þ n. An equivalent way to view S1  S3 is as the quotient of C2{0} by the Z-action that has n 2 Z acting to send a pair z ¼ (z1, z2) to the pair fn(z) ¼ (en z1, en z2). The pull-back of the basis {dz1, dz2} of T1,0C2 by jn is {endz1, endz2}, and so the jn pull-back of T1,0C2 is T1,0C2. This is to say that jn preserves the standard complex structure, j0, on C2{0}. As a consequence, this complex structure descends to the quotient to define one on S1  S3. Note that S1  S2n1 for any n > 2 gives a very analogous example. The latter can be viewed as the quotient of Cn{0} by the action of Z that has n 2 Z acting on the complex coordinates (z1, . . . , zn) to give (enz1, . . . , enzn). As it turns out, the product S2p1  S2q1 for any p, q  1 has a whole family of complex structures, no two of which are equivalent. The construction is due to Calabi and Eckmann: Let u 2 C denote a complex number with nonzero imaginary part. View C as an additive group, and let C act on (Cp{0})  (Cq{0}) as follows: An element z 2 C sends a pair (w1, w2) 2 (Cp{0})  (Cq{0}) to (ezw1, euzw2). Let XU denote the quotient space by this action. The result is a complex manifold by virtue of the fact that the action of C defined here is complex linear with respect to the coordinates (w1, w2) 2 Cp  Cq. This is to say that it commutes with the standard complex structure. It is easiest to see that the resulting space XU is diffeomorphic to S2p1  S2q1 when u ¼ i. In this case, there is a unique point in S2p1  S2q1 on the orbit of any given pair (w1, w2), this the point (w10 , w20 ) where w10 ¼ w1jw1j1jw2ji and w20 ¼ w2 jw2j1jw1ji. Example 17.10 An oriented manifold of dimension 2. Let  denote an oriented surface with metric g. As noted in Chapter 17.4, there is an associated, compatible almost complex structure. This is defined with respect to an oriented, orthonormal basis {e1, e2} by the rule je1 ¼ e2 and je2 ¼ e1. The volume form o ¼ e1 ∧ e2 is the Ka¨hler form. Example 17.11 The manifold CPn. I have defined CPn as S2nþ1/S1 where I am viewing S2nþ1 as the unit sphere in Cnþ1 and where S1 is viewed as the unit circle in C. A given element l 2 S1 sends a given (z1, . . . , znþ1) 2 S2nþ1 to (lz1, . . . , lznþ1). Let C*  C denote the multiplicative group of nonzero complex numbers, this, C{0}. As explained in Example 17.1, CPn is also (Cnþ1{0})/C*. The view of CPn as (Cnþ1{0})/C* is taken here as it allows for a direct view of the complex manifold structure: Recall from Example 17.1, an open cover of CPn is given by sets U ¼ {O k}1knþ1 where O k is the image in CPn of the set of points z ¼ (z1, . . . , znþ1) 2 C where zk 6¼ 0. The coordinate chart diffeomorphism from O 1 to Cn sends (z1, . . . , znþ1) to the point (z2/z1, z3/z1 . . . , znþ1/z1) 2 Cn. That from O 2 to Cn sends (z1, . . . , znþ1) to the point (z1/z2, z3/z2, . . . , znþ1/z2) 2 Cn, etc. The transition functions are holomorphic; for example, if j1: O 1 ! Cn denotes the coordinate chart map just described, and likewise j2: O 2 ! Cn, then transition function j2  j1 1 is

Appendix 17.1

Compatible almost complex structures

defined on the set (w1, w2, . . . , wn) in Cn where w1 is nonzero, and it sends such a point (w1, . . . , wn) to (1/w1, w2/w1, . . . , wn/w1). There is a special Ka¨hler metric for CPn. This metric is called the Fubini–Study metric, and it is denoted here by gFS. It is customary to define gFS by giving its pullback to Cnþ1{0} via the projection map to CPn. This pull-back is jzj2 dza

O

dza  jzj2za dza

O

zb dzb :

The metric on any given k 2 {1, . . . , n þ 1} version of O k pulls back to Cn via the inverse of the corresponding coordinate chart map jk as ðjk 1 Þ*ðgFS Þ ¼ ð1 þ jwj2 Þ1 dwj

O

 j  ð1 þ jwj2 Þ2 w  j dwj dw

O

 m: wm dw

Here, I have written it so that it is evident that gFS defines a Hermitian, positive definite form on T1,0O k. The metric on the underlying real vector space TCPn is obtained from this by writing each wk in terms of its real and imaginary part, thus as wk ¼ x2k1 þ ix2k. This done, then write dwk ¼ dx2k1 þ idx2k and insert into the expression above. I leave it as an exercise to verify using the coordinate chart transition functions that the expression written above for a given k 2 {1, . . . , n þ 1} is consistent with that defined by a different value of k on the overlap of the corresponding coordinate patches. The associated almost Ka¨hler form o pulls up to Cnþ1{0} as i 2 i jzj dza ∧ dza  jzj2za dza ∧ zb dzb : 2 2 On any given O k , this form pulls back via j1 k as i i 2 1  j  ð1 þ jwj2 Þ2 w  m:  j dwj ∧ wm dw ðj1 k Þ * oFS ¼ ð1 þ jwj Þ dwj ∧ dw 2 2 To see that it is closed, it is sufficient to note that i 2 1 j  w  j dwj ÞÞ: ðj1 k Þ * oFS ¼ dðð1 þ jwj Þ ðwj dw 4

Appendix 17.1

Compatible almost complex structures

Let o denote a 2-form on a 2n-dimensional manifold M whose n’th exterior power is nowhere zero. Suppose that j is an almost complex structure on TM. This almost complex structure is said to be o-compatible when the section N o(·, j(·)) of 2 T*M defines a Riemannian metric. This is to say that it defines a

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symmetric, positive definite inner product on TM. As noted in Chapter 17.6, every almost Ka¨hler form has compatible almost complex structures. What follows here is a proof of this assertion in the case when M is compact. The argument in general is almost the same but for the use of locally finite covers. The proof for compact M has five parts.

Part 1 Use on to define an orientation for M. Fix p 2 M and fix an oriented basis, v ¼ (v1, . . . , v2n) for TM. Define from v and o the 2n  2n, antisymmetric matrix Av whose (i, j) 2 {1, . . . , 2n} component is o(vi, vj). This matrix at p is nondegenerate, and so it has 2n eigenvalues, of which n are of the form i l with l a positive real number. The other n eigenvalues are the complex conjugates of the first n. It proves useful to require that these eigenvalues are distinct. If they are not for the given basis, then there is, in all cases, a change of basis that makes them distinct. I leave the proof of this to you as an exercise in linear algebra. With it understood now that the eigenvalues of Av are distinct, suppose that il is any given eigenvalue of Av with l > 0. Let e 2 C2n denote the corresponding eigenvector. Both the real and imaginary parts of e are nonzero as can be seen from the fact that the complex conjugate vector is an eigenvector with eigenvalue il. The point being that il 6¼ il and so e cannot be a complex multiple of its complex conjugate. Write e as e1 þ ie2 where e1 and e2 are in R2n. Write their respective components as {e1i}1i2n and {e2i}1i2n. P Use e 2 TMjp to denote the point 1i2n e1i vi and use e2 2 TMjp to denote P 1 i 1i2n e2 vi. Define an almost complex structure, j, on the span of {e1, e2} by the rule je1 ¼ e2 and je2 ¼ e1. Making this definition of j for each eigenspace defines j on the whole of TMjp as an almost complex structure. Now suppose that l, e1 and e2 are as just described. Given that e1 and e2 come from the il eigenvector of Av, it follows that o(e1, je1) ¼ l ¼ o(e2, je2) and o(e1, je2) ¼ 0 ¼ o(e2, je1). Thus, the bilinear form given by o(·, j(·)) defines a Riemannian metric on the span of {e1, e2}. Suppose that l0 6¼ l, l0 is positive and il0 is an eigenvalue of Av. Let e0 denote the correponding eigenvector. Write it in terms of real vectors in R2n as e10 þ ie20 . As l0 6¼ l, one has o(e1, je10 ) ¼ 0 ¼ o(e1, je20 ) and likewise o(e2, je10 ) ¼ o(e2, je10 ). Reversing the roles of the prime and unprimed components shows that the span of {e1, e2} is orthogonal with respect to o(·, j(·)) to the span of {e10 , e20 }. Granted this, it follows that j as just defined is an o-compatible almost complex structure at p.

Appendix 17.1

Compatible almost complex structures

Part 2 The task for this part of the argument is to extend this almost complex structure to an open neighborhood of p. This is done as follows: First extend the basis v to a neighborhood of p. This is a straightforward procedure that can be done, for example, by fixing an open set containing p which comes with an isomorphism from TM to the product bundle. With this isomorphism to the product bundle understood, extend the basis as the constant basis. Given such an extension, any given point p0 in a neighborhood of p has a corresponding antisymmetric matrix, Avjp0 . If the eigenvalues are distinct at p, they will be distinct at each such point in some small neighborhood of p. This said, the corresponding eigenspaces vary near p so as to define a set of n complex line bundles. To elaborate, let U  M denote such a neighborhood. L L L Then the eigenspaces define a decomposition TMCjU ¼ ( 1an Ea) ( 1an  a) where each Ea is a complex line bundle. Define j on TMjU so that its (1, 0) E L tangent space in TMCjU is the summand ( 1an Ea). See Chapter 6.3 if your memory fails you with regards to this definition of an almost complex structure. In any event, it follows from the observations from Part 1 that o(·, j(·)) is a Riemannian metric on TMjU.

Part 3 Part 5 explains how to obtain an o-compatible almost complex structure for the whole of TM from those constructed in Part 3 on small open sets. As you might guess from the discussion at the outset of Chapter 7, the construction involves partitions of unity for an open cover. However, the story here is not as simple because the sum of two almost complex structures gives an endomorphism of TM whose square is not, in general, equal to minus the identity. This part and Part 4 set up the machinery to deal with this complication. To start, consider an open set U  M where TM has a metric compatible, almost complex structure, j. Take a smaller open set if necessary so as to guarantee that there is an orthonormal basis of sections of TMjU for the metric g(·, ·) ¼ o(·, j(·)). Use the metric to identify TMjU with T*MjU and use the basis to identify both isometrically with U  Rn. With this identification understood, both o and j appear as maps from U to the space M(2n; R) of 2n  2n matrices. Use W to denote the former and J to denote the latter. This is to say that the (i, j) component of W is o(ei, ej) where ei and ej denote the i’th and j’th elements of the chosen orthonormal basis. Meanwhile, the (i, j) component of J is ei(Jej) where here ei denotes the i’th component of the dual basis for T*M. Since the identification between TMjU and T*MjU is isometric, the identification sends the metric to the identity matrix i. This implies that W ¼ J1 at each point of U.

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Meanwhile, J2 ¼ i, and so J1 ¼ J at each point of U. Moreover, JT ¼ J as j and hence J acts orthogonally. Now suppose that j0 is a second, o-compatible almost complex structure on TMjU. Let J0 denote the corresponding map from U to the space 2n  2n matrix. This map is also defined using the orthonormal frames as defined by the metric g for TMjU and T*MjU. To say that j0 is o-orthogonal means neither more nor less than the following: The matrix WJ0 is a symmetric, positive definite matrix at each point of U. Let H denote the latter. Since W ¼ J1, this means that J0 ¼ JH. If (J0 )2 is to equal i, then JH ¼ (JH)1 ¼ H1J1. Multiplying both sides of this equation by J finds that H ¼ JH1J1. This understood, the task for what follows is to identify the set of maps from U to the space of symmetric, positive definite, 2n  2n matrices that obey H ¼ JH1J1 at each point. This task is accomplished in the next part of the Appendix.

Part 4 This part of the appendix identifies the space of maps from U to the positive definite, 2n  2n symmetric matrices that obey H ¼ JH1J1 with a space of maps from U to a certain linear subspace in the space of 2n  2n matrices. The key point for use later is that the subspace in question is linear. The identification is given in three steps. Step 1: Fix a positive integer m, and let Symþ(m; R)  M(m; R) denote the open subspace of positive definite, symmetric m  m matrices. In the application to the problem at hand, m will be equal to 2n. Let Sym(m; R) denote the space of symmetric, m  m matrices. The lemma below refers to the exponential map that sends M(n; R) to Gl(n; R). The latter was introduced in Chapter 5.5; it sends a matrix m to em with em defined by its power series: em ¼ i þ m þ 12 m2 þ 3!1 m3 þ : : : . Lemma 17.1 The exponential map m ! em restricts to Sym(m; R) as a diffeomorphism onto Symþ(m; R). Proof of Lemma 17.1 Look at the power series definition of em to see that each term is symmetric when m is symmetric. To see that em is positive definite, it is enough to note that em can be written as em/2em/2 which is the square of a symmetric matrix. Hence, all the eigenvalues of em are squares and so all are positive. None are zero because the exponential map lands in Gl(m; R). To see that this map is onto Symþ(m; R), remark that any given positive definite, symmetric matrix h can be written as odo1 where o is orthogonal and d is a diagonal matrix with positive entries. This said, let ln(d) denote the diagonal matrix whose entry on any given diagonal is the natural log of the corresponding entry of d. Set m ¼ o ln(d)o1. A look at the power series expansion shows that h ¼ em.

Appendix 17.1

Compatible almost complex structures

The map m ! em is also 1–1 as can be seen by the following argument: 0 Suppose that em ¼ em . Since m is symmetric, it too can be written as odo1 with d a diagonal matrix. This implies that em ¼ oedo1 and so the eigenvalues of em are the exponentials of those of m. It follows from this that m and m0 have the same set of eigenvalues and corresponding eigenvalues have the same multiplicity. This implies that m0 can be written as umu1 with u a diagonal 0 0 matrix. But if the latter is true, then em ¼ uem u1, and this can happen only if u preserves the eigenspaces of m. This is to say that umu ¼ m and so m0 ¼ m. The next point to make is that the differential of the map m ! em on Sym(m; R) defines an isomorphism at each point. To see why this is, fix m, q 2 Sym(m; R) and consider for t 2 (0, 1) but small the map t ! emþtq. A look at the derivative of this map at t ¼ 0 finds the latter given by ð1

eð1sÞm qesm ds:

0

This is m’s image via the differential at m of the exponential map. To see when this is zero, write m ¼ o d o1 with d a diagonal matrix and d an orthogonal matrix. What is written above is then 01 1 ð   o@ eð1sÞd o1 om esd dsAo1 : 0

This can vanish only if the integral is zero. To see when such is the case, introduce zij denote the (i, j) entry of the matrix o1q o. Let di denote the i’th diagonal entry of d. Then the (i, j) entry of the matrix that appears in the integrand above is equal to eð1sÞdi þsdj Zij : In particular, the integral of this with respect to s is zero if and only if zij ¼ 0. As just argued, the map m ! em is 1–1 and onto; and its differential is everywhere an isomorphism. This implies that it is a diffeomorphism. Step 2: Suppose that H is a map from U to the space Symþ(2n; R). It follows from the lemma in Step 1 that this map can be written in a unique way as H ¼ eQ where Q is a smooth map from U to Sym(2n; R). The inverse matrix H1 is eQ. This being the case, then the condition H ¼ JH1J1 requires that JQJ1 ¼ Q. Such is the case if and only if Q can be written as Q ¼ V  J1VJ with V a smooth map from U to Sym(2n; R). To put this in perspective, remark that the map Q ! JQJ1 defines a linear map from Sym(2n; R). This is because J1 ¼ J and J ¼ JT, so this is the same as the map sending Q to JQJT. Let #: Sym(2n; R) ! Sym(2n; R) denote this map

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17 : Complex manifolds

Q ! JQJT. Since J2 ¼ 1, this map # obeys #2 ¼ 1, so its eigenvalues are 1. Of interest here is the 1 eigenspace, because if #(Q) ¼ Q, then Q ¼ JQJ1, and conversely if Q ¼ JQJ1 then Q is an eigenvector of # with eigenvalue 1. The dimension of this eigenspace is the same as that of the vector space (over R) of Hermitian, n  n matrices. This follows from what is said in the final paragraph of the proof of Lemma 6.4. Thus, this 1 eigenspace has dimension n(n þ 1). Denote this 1 eigenspace by S(2n; R). Step 3: To summarize: A map H from U to Symþ(2n; R) that obeys H ¼ JH1J1 can be written in a unique fashion as eQ where Q is a smooth map from U to S(2n; R). Conversely, if Q is such a smooth map, then H ¼ eQ is a map from U to Symþ(2n; R) that obeys H ¼ JH1J1.

Part 5





It follows from what was said in Part 3 that M has a finite cover U such that each U 2 U comes with an o-compatible almost complex structure on TM. This cover should be chosen so as to have the following two additional features: First, each U  U comes with a diffeomorphism, jU, to the interior of the ball of radius 3 centered at the origin in R2n. Note that this implies that TMjU is isomorphic to the product bundle U  R2n. To state the second, introduce B1  R2n to denote the ball of radius 1 centered at the origin. Here is the second requirement: The collection {j1 U (B1)}U2U is also an open cover of M. Let N denote the number of sets in this cover. Label these sets as {Uk}1kN. For each k 2 {1, . . . , N}, let jk to denote the U ¼ Uk version of jU. Likewise, let jk denote the o-compatible, almost complex structure given on Uk. For each k 2 {1, . . . , N}, use B2k/N  R2n to denote the ball of radius 2  k/N centered at the origin. Introduce M(k)  M to denote 1mk jk1(B2k/N). Note that M(N) ¼ M because the collection {jk1(B1)}1kN covers M. The construction of the desired o compatible almost complex structure will be done in a step-by-step fashion. The k’th step will result in an o-compatible almost complex structure over M(k). The latter will be denoted in what follows by j(k). To start, consider M(1). This is a subset of U1. The desired o-compatible almost complex structure on M(1) is j(1) ¼ j1. Now to proceed by induction, suppose that k 2 {1, 2, . . . , N  1} and that j(k) has been defined on M(k). What follows defines j(kþ1) on M(kþ1). There are two cases to consider. In the first case, Ukþ1 is disjoint from M(k). In 1 this case, M(kþ1) is the disjoint union of jkþ1 (B2(kþ1)/N) with the subset (k) (kþ1) 1 to equal jkþ1 in 1mk jk (B2(kþ1)/N) of M . This understood, set j (k) 1 jkþ1 (B2(kþ1)/N) and equal to j on the component of M(kþ1) in M(k). Now suppose that Ukþ1 M(k) is not empty. Denote this subset of Ukþ1 by Vk. Let Vkþ1  Ukþ1 denote Ukþ1 M(kþ1). Note that Vkþ1  Vk and that the closure ¨

¨

266

Additional reading

of Vkþ1 in M is disjoint from that of Ukþ1  Vk in M. This is an important observation so keep it in mind. Define j(kþ1) on the complement in M(kþ1) of VkVkþ1 to equal j(k). To define (kþ1) j on Ukþ1 M(kþ1), first introduce the metric gkþ1 ¼ o(·, jkþ1(·)) for TUkþ1. As TUkþ1 is isomorphic to the product bundle, there exists an orthonormal basis of sections for this metric. Use this basis as in Part 3 to identify jkþ1 with a map, J, from Ukþ1 to the space of 2n  2n matrices. The desired almost complex structure j(kþ1) will also be given over the part of M(kþ1) in Ukþ1 as a map, J*, to this same space of matrices. To define J*, use the chosen gkþ1-orthonormal frame for TUkþ1 to identify j(k) on Vk with a map, J0 , from Vk to the space of 2n  2n matrices. As seen in Part 4, the map J0 is such that H ¼ J1J0 has the form eQ where Q is a map from Vk to S(2n; R). This is to say that J0 ¼ JeQ. Given that the closure of Vkþ1 is disjoint from that of Ukþ1Vk, there is a strictly positive lower bound to the distance between any point in the set jkþ1(Vkþ1) B2 and B2jkþ1(Vk). As a consequence, the constructions from Appendix 1.2 can be used to obtain a smooth function on B2  R2n that is equal to 1 on jkþ1(Vkþ1) and equal to zero on B2jkþ1(Vk). Use y to denote the composition of this function with the map jkþ1. Then set J* ¼ JeyQ. Note that using J* to define j(kþ1) on M(kþ1) Ukþ1 gives an almost complex structure that agrees with the definition given previously for j(kþ1) on the set M(kþ1)(M(kþ1) (VkVkþ1); this is because y Q is equal to Q on Vkþ1. ¨

¨

¨

¨

Additional reading  Complex Geometry: An Introduction, Daniel Huybrechts, Springer, 2004.  Lectures on Ka¨hler Geometry, Andrei Moroianu, Cambridge University Press, 2007.  Foundations of Differential Geometry, Volume II, Shoshichi Kobayashi and Katumi Nomizu, Wiley Interscience, 1996.  Algebraic Curves and Riemann Surfaces, Rick Miranda, American Mathematical Society, 1995.  Complex Manifolds, James Morrow and Kunihiko Kodaira, American Mathematical Society, 2006.  Complex Manifolds without Potential Theory, Shing-shen Chern, Springer, 1979.  Introduction to Symplectic Topology, Dusa McDuff and Dietmar Salamon, Oxford University Press, 1999.

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18

Holomorphic submanifolds, holomorphic sections and curvature

This chapter describes how complex manifolds and Ka¨hler manifolds can appear as submanifolds in a larger complex or Ka¨hler manifold.

18.1

Holomorphic submanifolds of a complex manifold

The examples here concern holomorphic submanifolds of a complex manifold. To set the stage, suppose that N is a complex manifold. Let j denote the almost complex structure on N that comes from its complex manifold structure. A submanifold M  N is said to be holomorphic if j along M maps the subbundle TN  TMjN to itself. Thus, j maps any given tangent vector to M to some other tangent vector to M. Assume that M has this property. Then jjM defines an endomorphism of TM with square 1, so an almost complex structure, jM, on TM. Moreover, the Nijenhuis tensor of this almost complex structure is zero by virtue of the fact that it is zero for j. To see this, remember that the Nijenhuis tensor vanishes if and only if the commutator of two sections of T1,0M is also a section of T1,0M. Here, sections of T1,0M are viewed as C-valued tangent L vectors on M and TMC is written as T1,0M T0,1M. This understood, remark that T1,0M  T1,0NjM. Thus, the commutator of two sections of T1,0M is, a priori, in T1,0NjM; and thus in T1,0M since commutators of vectors tangent to M are tangent to M, whether complex or not. Given that the Nijenhuis tensor of jM is zero, it follows from the Newlander–Nirenberg theorem (Theorem 17.5) that jM comes from a complex manifold structure on M. Now suppose that g is a Ka¨hler metric on M. Then g’s restriction to TM  TNjM defines a metric, gM, on TM, and this is compatible with jM since they are restrictions of g and j to a subspace in TNjM. Let o denote the

18.2

Holomorphic submanifolds of projective spaces

associated almost Ka¨hler form o on N. Then the associated almost Ka¨hler form, oM, for the pair gM and jM is the restriction of o as a bilinear form on TM  TNjM. This is just the pull-back via the inclusion map M ! N of o. It follows from this that if o is closed, then oM is also closed. These last observations are summarized by the following: Proposition 18.1 Let N be a complex manifold and M a holomorphic submanifold of real dimension at least 2. Then M also has the structure of a complex manifold. Moreover, if g is a Ka¨hler metric for N, then the restriction of g to the tangent space of M defines a Ka¨hler metric for M.

18.2

Holomorphic submanifolds of projective spaces

Chapter 17.3 explains how CPn can be viewed as a complex manifold, and Chapter 17.7 describes its associated Ka¨hler form. A plethora of holomorphic Ka¨hler submanifolds can be found inside of CPn. What follows describes an archetype example: Let z ¼ (z1, . . . , znþ1) denote the coordinates for Cnþ1. Let h: Cnþ1 ! C denote a polynomial in the coordinates (z1, . . . , znþ1) that is homogeneous in the sense that there exists a positive integer p such that h(lz) ¼ lp h(z) for all l 2 C. Here, lz is the point with coordinates (lz1, . . . , lznþ1). Let Zh ¼ {z 2 Cnþ1{0} : f (z) ¼ 0}. This subset is invariant under the action of C* on Cnþ1 whose quotient gives CPn. Use Mh  CPn to denote the image of Zh under the projection. The proposition that follows refers to the holomorphic differential of a C-valued function. Here is the definition: Let m  1, let U  Cm denote an open set, and let f : U ! C denote a smooth function. The holomorphic differential of f is the projection of dh onto the span of the 1-forms {dwi}1im. This is usually written as @f . To elaborate, @f ¼ ∑1im

@f dwi @wi

@ where dwi ¼ dx2i1 þ idx2i and @w ¼ 12 ð@x@2i1  i @x@2i Þ. Here, wi ¼ x2i1 þ ix2i is i written in terms of its real and imaginary parts. Note that a function f : U ! C is holomorphic as defined in Chapter 17.1 when @f df ¼ @f . This is to say that @@fw ¼ 12 ð@x@f2i1  i @x Þ ¼ 0 for all i. If f is written as f1 þ if2 2i i

with f1 and f2 both ordinary R-valued functions, then this says that f1 and f2 obey the Cauchy–Riemann equations for each of the coordinate pairs (x2i1, x2i): @f 1 @f ¼ 2 @x2i1 @x2i

and

@f 2 @f ¼ 1 : @x2i1 @x2i

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Polynomials of the complex coordinates on Cnþ1 are examples of holomorphic functions on the whole of Cnþ1. Proposition 18.2 Suppose that h is a homogeneous polynomial of the coordinates on Cnþ1 whose holomorphic differential is nowhere zero along h1(0)  Cnþ1{0}. Then the image, Mh  CPn of h1(0) is a compact, complex submanifold of CPn of real dimension 2n  2. Moreover, the complex manifold structure coming from CPn is such that the induced metric from the Fubini– Study metric on CPn makes Mh a Ka¨hler manifold. This proposition about holomorphic submanifolds of CPn is proved momentarily. Examples for the case n ¼ 2 are the polynomials h ¼ zp1 þ zp2 þ zp3 . The corresponding submanifold of CP2 is 2-dimensional, and so a surface. The genus of this surface is 12 ¼ ðp2  3p þ 2Þ. Examples in the case n ¼ 3 are h ¼ zp1 þ zp2 þ zp3 þ zp4 . These are all simply connected manifolds of real dimension equal to 4. The proof of the proposition requires the following preliminary lemma. Lemma 18.3 Let U  Cm denote an open set and let f : U ! C denote a holomorphic function. Then f 1(0) is a smooth, (2m  2)-real dimensional submanifold near any point of f 1(0) where @f 6¼ 0. Proof of Lemma 18.3 View f as a map from R2n to R2. Viewed in this light, the implicit function theorem from Chapter 1.2 tells us that f 1(0) is a submanifold of U near any point where the corresponding matrix of partial derivatives is surjective. This matrix has 2 columns and 2n rows, and its entries are the partial derivatives of the components of f . To see that this matrix is surjective under @f the assumptions of the lemma, suppose for the sake of argument that @z is 1 nonzero at some point where f is zero. Then it is nonzero in a neighborhood of this point. Keeping this in mind, write f as f1 þ i f2 with f1 and f2 real. Also, write z1 ¼ x1 þ ix2 with x1 and x2 real. The 2  2 minor of the differential of f (as a map from R2nþ1 to R2) corresponding to the (x1, x2) coordinates is   @1 f 1 @1 f 2 @2 f 1 @2 f 2 where @ 1 and @ 2 here are the respective partial derivatives with respect to x1 and x2. This understood, it is enough to verify that the determinant of this @f 2 matrix is nonzero. This determinant is j@z j when @@fz ¼ 0. 1 1 Note that the proof of this lemma also establishes the following: Lemma 18.4 Let U  Cm denote an open set and let f : U ! C denote a holomorphic function. The point 0 2 C is a regular value of f if and only if @f 6¼ 0 along f 1(0).

18.3

Proof of Proposition 18.2, about holomorphic submanifolds of CPn

Recall that a regular value of a map, f : Rp ! Rq is a point y 2 Rq with the following property: Let x denote any point with f (p) ¼ y. Then the matrix with p columns and q rows whose entries are the partial derivatives of the coordinates of f is surjective. Proof of Lemma 18.4 As seen in the proof of Lemma 18.3, the vanishing of @f at any given point p 2 U is equivalent to the assertion that the differential of f is zero at that point. Here, I view   f1 f ¼ f2 as a map from an open set in Cn ¼ R2n to C ¼ R2 so as to view its differential as a matrix with 2n columns and 2 rows whose entries are partial derivatives of f . By the way, Lemma 18.4 and Sard’s theorem (Theorem 1.3) together supply the following additional conclusion: Corollary 18.5 Let U  Cm denote an open set and let f : U ! C denote a holomorphic function. Then almost all values c 2 C are regular values of f. As a consequence, f 1(c) is a holomorphic submanifold of U for almost all values c 2 C. Recall that Sard’s theorem says the following about a given map from Rp to Rq: The set of nonregular values of the map has zero measure. Thus, the set of regular values in a given ball has full measure. This is what Corollary 18.5 means by the term almost all. To obtain the corollary, apply Lemma 18.4 to the function f  c with c being a regular value of f .

18.3

Proof of Proposition 18.2, about holomorphic submanifolds of CPn

Lemma 18.3 guarantees that f 1(0) is a smooth submanifold of Cnþ1{0}. Chapter 17.3 describes an open cover, {O k}1knþ1 of coordinate charts, each diffeomorphic to Cn via a diffeomorphism that makes for holomorphic coordinate transition functions. By way of a reminder, the coordinate chart O k consists of the set of complex lines through the origin in Cnþ1 on which the k’th coordinate zk 6¼ 0. The coordinate chart map fk is defined so that the line spanned by a point (z1, . . . , znþ1) 2 Cnþ1{0} with zk 6¼ 0 is sent to the point in Cn whose coordinates are (z1/zk, . . . , zk1/zk, zkþ1/zk, . . . , zn/zk). A calculation, done in the coordinate charts O k using the coordinates given by jk, shows that Mh O k is a smooth submanifold of Ok. For example, j1(Mh O 1) is the submanifold of Cn given by

271

¨

¨

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hð1; w1 ; . . . ; wn Þ ¼ 0: To see if this locus is a submanifold, it is enough to verify that at least one of the @h is nonzero at any given point along this locus. To see that partial derivatives @w m they can’t all vanish, remark that the inverse image of O k in Cnþ1 has coordinates (u, w1, . . . , wn) 2 Cnþ1 which are related to the z-coordinates by the rule z1 ¼ u, z2 ¼ uw1, . . . , znþ1 ¼ uwn. In terms of the coordinates (u, w1, . . . , wn), the function h appears as hðz1 ; z2 ; . . . ; znþ1 Þ ¼ up hð1; w1 ; . . . ; w2 Þ: If all of the partial derivatives of the function w ! h(1, w1, . . . , wn) vanish where this function is zero, then every partial derivative of the function (u, w) ! up h(1, w1, . . . , wn) is also zero, so every partial derivative of z ! h(z1, . . . , znþ1) is zero too. But this is forbidden by the assumption that @h is nonzero on Cnþ1{0} where h ¼ 0. To see that Mh is a holomorphic submanifold, it is sufficient to verify that the almost complex structure on Cn preserves the tangent space to the zero locus of the function w ! h(1, w1, . . . , wn). To start this task, suppose for the sake of @h argument that @w is nonzero at a given point on this locus. Then it will be 1 nonzero at all nearby points. This understood, it follows that the tangent space to the h ¼ 0 locus in a neighborhood of the given point is spanned by the real and imaginary parts of the vectors ( )  1  @ @h @h @  : @wj @wj @w1 @w1 2jn

These vectors are mapped to themselves by j. Note that the argument just given proves the following general fact: Proposition 18.6 Let U  Cn denote an open set and let f : U ! C denote a function that is holomorphic with respect to the coordinate functions of Cn. The f 1(0) is a holomorphic submanifold of Cn on a neighborhood of any point P @f dwj is nonzero. where the holomorphic differential 1jn @w j

18.4

The curvature of a Ka¨hler metric

Suppose that M is a complex manifold and g is a metric that makes the complex structure Ka¨hler. As seen in Chapter 17.9, the Ka¨hler form o and J are both covariantly constant with respect to the Levi-Civita connection. To see what this implies, remember that if J is covariantly constant, then the Levi-Civita N covariant derivative takes a section of T1,0M to a section of T1,0M T*MC.

18.4

The curvature of a Ka¨hler metric

This is to say that it defines a covariant derivative for T1,0M. By way of reminder as to how this comes about, I can choose an orthonormal basis, {eb}1b2n, of sections for T*M on a neighborhood of a given set such that Je2k1 ¼ e2k and Je2k ¼ e2k1. The action of  is determined by its action N on these basis 1-forms. This I write as eb ¼ ea W ab where each W ab is a ab ba 1-form, and where W ¼ W . This is to say that the collection {W ab} define a 2n  2n antisymmetric matrix of 1-form. (Recall that the vector space of 2n  2n, antisymmetric matrices is the Lie algebra of SO(2n).) Meanwhile {uk ¼ e2k1 þ ie2k}1kn is a basis for the dual space T1,0M  TMC to T1,0M. Given that J ¼ J, one finds that uk ¼ um

O

Bmk

where each Bmk is a section of T*MC, and these are such that Bkm ¼ Bmk . Here,   1  ð2m1Þð2k1Þ Bmk ¼ þ Wð2mÞð2kÞ þ i Wð2m1Þð2kÞ  Wð2mÞð2k1Þ : W 2 Note that  and J commute if and only if W(2m1)(2k1) ¼ W(2m)(2k) and W(2m1)(2k) ¼ W(2m)(2k1). The collection {Bmk} defines an n  n, antiHermitian matrix of 1-forms. (Recall that the vector space of n  n, anti-Hermitian matrices is the Lie algebra of U(n).) The induced covariant derivative  on T1,0M has a curvature 2-form. When written near the given point used above in terms of the frame {uk}, this curvature 2-form is an n  n, anti-Hermitian matrix of 2-forms; its components being {(F )mk}. It is defined so that O  mk d uk ¼ um : F The components {(F )mk} can be expressed in terms of the components, {Rabcd}, of the Riemann curvature tensor from Chapter 15.7 as follows: Write N deb ¼ ea (F)ab where (F)ab ¼ 12 Rabcdec∧ed. Then   mk 1  F ðF Þð2m1Þð2k1Þ þ ðF Þð2mÞð2kÞ þ iðF Þð2m1Þð2kÞ  ðF Þð2mÞð2k1Þ : ¼ 2 Note that the condition that  and J commute implies that (F)(2m1)(2k1) ¼ (F)(2m)(2k) and also (F)(2m1)(2k) ¼ (F)(2m)(2k1). The formula above guarantees that ðF  Þmk ¼ ðF  Þkm and so the collection {(F )mk} does indeed supply the entries of an n  n, anti-Hermitian matrix of 2-forms. The proposition that follows describes two important features of F . Two remarks are needed to set the stage. Here is the first: The End(T1,0)-valued 2-form F  can be decomposed as (F )2,0 þ (F )1,1 þ (F )0,2 where the

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subscripts correspond to the decomposition of End(T1,0M) the summands EndðT1;0 MÞ

O

ðT2;0 M

M

T1;1 M

M

N

∧2 T*MC into

T0;2 MÞ:

The proposition that follows asserts that F  has only a 1–1 component. The second remark concerns the trace of F , this now a closed 2-form on M. Recall from Chapter 14.5 that 2pi tr(F ) represents the first Chern class of T1,0M in H2De Rham(M). The proposition writes this form in terms of the Ricci tensor from Chapter 15.7. Proposition 18.7 The (0,2) and (2,0) components of F  are zero. Meanwhile, the trace of F  is determined by the Ricci tensor of the metric g by 1 trðF  Þ ¼ ∑1mn ðF Þ mm ¼  ∑1m;kn ðRicð2mÞð2kÞ Þ þ ðRicð2m1Þð2k1Þ Þum ∧u k: 4 {

Proof of Proposition 18.7 The first point to make is that (F )0,2 ¼ (F )2,0 where the operation { here denotes the Hermitian adjoint. This is induced from the R-linear automorphism of the bundle End(T1,0M) that acts to send a given m 2 End(T1,0M) to the adjoint m{. The latter automorphism is then extended to N 2 give an automorphism of End(T1,0M) N ∧ T*MC by defining it on the reducN ible elements. In particular, it acts on m k to give m{ C. To see that (F )0,2 ¼ 0 near any given point, use a frame {uk}1kn as N introduced above. The formula duk ¼ um F mk means that d d uk ¼ ðF Þmk ∧um : This must be zero because  is torsion free. The expression on the right is a C-valued 3-form on M, and so has components of type (3, 0), (2, 1), (1, 2) and (0, 3). The latter is evidently zero. Meanwhile, the (1, 2) component is ((F )mk)0,2 ∧ um, and this is zero if and only if ((F )mk)0,2 ¼ 0 for all m and k. To prove the claim about the trace of F , write the trace of F  in terms of F and so find it equal to i trðF  Þ¼ ∑1mn ððF Þð2m1Þð2mÞ  ðF Þð2mÞð2m1Þ ¼ i∑1mn ðF Þð2m1Þð2mÞ: 2 The right-most equality uses the fact mentioned earlier that (F)(2k1)(2j) ¼ (F)(2k)(2j1). To continue, write ðF Þab ¼ 12 Rabcd ec ∧ ed and then write e2m1 as 1 m m Þ and likewise write e2m as 2i1 ðum   um Þ. Use these identifications to write 2 ðu þ u trðF  Þ ¼

1 ðRð2m1Þð2mÞð2k1Þð2jÞ Þuk ∧ uj : 2

This uses the identities mentioned earlier, (F)(2k1)(2j1) ¼ (F)(2k)(2j) and (F)(2k1)(2j) ¼ (F)(2k)(2j1), in their guise as identities for Rabcd. This is to say that

18.5

Curvature with no (0, 2) part

Rð2xÞð2zÞcd ¼ Rð2x1Þð2z1Þcd and Rð2x1Þð2zÞcd ¼ Rð2xÞð2z1Þcd for any x; z 2 f1; . . . ; ng : A clever use of the identities Rabcd þ Racdb þ Radbc ¼ 0 and Rabcd ¼ Rcdab from Chapter 15.7 with the identities just written will identify the expression above for tr(F ) with the expression given in the proposition.

18.5

Curvature with no (0, 2) part

Suppose that M is a complex manifold and p: E ! M is a complex vector bundle. Let  denote a covariant derivative for sections of E. The associated N 2 curvature 2-form, F, is a priori a section of End(E) ∧ TMC. As such, it has (2, 0), (1, 1) and (0, 2) components with respect to the decomposition of L L ∧2 T*M as T 2,0M T1,1M T 0,2M. As is explained momentarily, a covariant derivative whose curvature has no (0, 2) part is especially significant. Note that if E has a Hermitian metric and  is metric compatible, then (F)0,2 ¼ (F{ )2,0 and so a metric compatible covariant derivative whose curvature has no (0, 2) part is purely of type (1, 1). The significance of the vanishing of (F)0,2 stems from the following proposition: Proposition 18.8 Suppose that M is a complex manifold and p: E ! M is a complex vector bundle of rank q with covariant derivative, , whose curvature 2-form has no (0, 2) part. Then E has the structure of a complex manifold with the property that each fiber of p is a holomorphic submanifold, and as such a copy of Cn with its standard complex structure. In particular, addition of a given vector to other vectors in the same fiber, or multiplication of a vector by a complex number both define holomorphic maps from the fiber to itself. The assertion that a fiber of E is a complex manifold means the following: Let JE: TE ! TE denote the almost complex structure on E given by its complex manifold structure. Then JE maps the kernel of p* to itself, and so induces an almost complex structure on any given fiber. This makes the fiber into a complex manifold. The proposition asserts that this complex manifold structure is just that of Cn. A bundle E with a covariant derivative whose curvature has no (0, 2) part is said to be a holomorphic vector bundle over M. Holomorphic vector bundles E and E0 are said to be holomorphically equivalent if there is a bundle isomorphism j: E ! E0 whose differential intertwines the respective almost complex structures.

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For example, if M is a Ka¨hler manifold, then the covariant derivative on the bundle T1,0M has curvature of type (1, 1). As a consequence, the total space of T1,0M is a complex manifold of the sort that is described by the proposition. Proof of Proposition 18.8 The first step is to define the almost complex structure JE on TE. This is done by specifying T1,0E. To start, fix attention on an open set U  M where there is an isomorphism j: EjU ! U  Cm. Such an isomorphism identifies TE over EjU with the tangent space to U  Cm, and T*E over EjU with T*(U  Cm). Use j to write a section of EjU in terms of a map from U to Cm. This is to say that if s is a section, then j(s) can be written as the section x ! (x, sU(x)) where sU is a map from U to Cm. By the same token, write N j(s) as a section x ! (x, (s)U) of (U  Cm) T*M and write ðsU Þ ¼ dsU þ asU where a is an m  m, complex valued matrix of 1-forms defined over U. Write the components of this matrix as {aab}1a,bm so that each component is a 1-form. These component 1-forms enter the definition of T1,0E as follows: This vector subspace over U is the pull-back via the identification j: EjU ! U  Cm of the vector space T1;0 MjU

M

Spanfdza þ aab zb g1am:

Here, {za}1am are the standard complex coordinate functions for Cm. Also, repeated indices are to be summed. Note that the definition given above is consistent with regards to the analogous definition made for EjU0 when U0  M intersects U. This is guaranteed by the manner by which the corresponding matrix of 1-forms a0 that defines  in U0 is related to the matrix of 1-forms a that defines  in U. See Chapter 11.2 if you are not remembering the relevant formula. With T1,0E defined as above, the next task is to compute its Nijenhuis tensor. Since M is assumed to be complex, it is enough to verify its vanishing over U by seeing whether it maps the forms {ya ¼ dza þ aabzb}1am to zero. To see that such is the case, note that dya ¼ dðdza þ aab zb Þ ¼ daab zb  aab ∧ dzb ¼ ðdaab þ aac ∧ acb Þzb  aac ∧ ðdzc þ acb zb Þ: The expression on the far right here can be written so as to identify dya ¼ ðF Þab zb  aac ∧ yc : The (0, 2) part of this ((F)ab)0,2zb. This is to say that the Nijenhuis tensor vanishes if and only if F has zero (0, 2) part.

18.6

Holomorphic sections

Granted that F has no (0, 2) part, the Newlander–Nirenberg theorem (Theorem 17.5) tells us that E has the structure of a complex manifold. To see that if the fibers of p are holomorphic submanifolds, it is enough to verify that JE maps the the tangent space of any given fiber to itself. This follows from the fact that the vector fields f@z@ a g1am span a subbundle in T1,0E as each is annihilated by all 1-forms in T0,1E.

18.6

Holomorphic sections

Let M and M0 denote even dimensional manifolds with respective almost complex structures j and j0 . A map f : M ! M0 is said to be pseudoholomorphic when its differential intertwines the almost complex structures. Thus f j ¼ j0 f . If J and J0 come from honest complex structures, then f is said to be * * holomorphic. This terminology is used for the following reason: Let U  M denote an open set and suppose that j: U ! Cn is an embedding onto an open set O whose differential sends j to the standard complex structure j0 on Cn. This is to say that j j ¼ j0j . Suppose that U is such that f (U)  U0 where * * 0 U0 is an open set M0 with an analogous embedding j0 : U0 ! Cn onto an open set O 0 . Let f U ¼ j0 f j: O ! O 0 . Write the coordinates of f U as (f1, . . . , fn0 ). If the map f is holomorphic, then each of the coordinate function fk is a k holomorphic function of the coordinates on O  Cn. This is to say that @f @zj for all j 2 {1, . . . , n}. Conversely, if the coordinates of f U are holomorphic, and this is true for all charts U from a cover of M, then f is holomorphic. Note that if f embeds M in M0 and is holomorphic, then f (M) is a holomorphic submanifold of M0 . An important example occurs in the following context: Let M denote a complex manifold and p: E ! M denote a complex vector bundle with a covariant derivative  whose curvature has vanishing (0, 2) part. As indicated above, E has a complex structure. When viewed as a complex manifold, the map p is holomorphic in the sense above. To see this, go to an open set U where there is an isomorphism j: EjU ! U  Cm. The map j identifies T1,0E over EjU with T1;0 MjU

M

Spanfdza þ aab zb g1am:

ð*Þ

The differential of p acts on this space by projecting to the T1,0U summand. This implies that p is holomorphic since it intertwines the complex structure endomorphism of TE with that of TM. What follows describes another example. View a section, s, of E as a map from the complex manifold M to the complex manifold E. Viewed in this

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way, a section defines a holomorphic map between these two complex maniN folds if and only if its covariant derivative, s, as a section of E T*MC has N no part of type (0, 1). By way of reminder, s is a section of E T*MC and N 1,0 N 0,1 L so can be viewed as a section of the direct sum (E T M) (E T M). The components of s with respect to this splitting are written as (@ s, @s). The claim is that the section s defines a holomorphic map when @s ¼ 0. To see this, write f(s) as a map x ! (x, sU(x)) 2 U  Cm. The pull-back via this map of a form u in the T1,0U summand in (*) is u, and so it lies in the T1,0M summand of T*MC. Meanwhile, the pull-back of one of the forms dza þ aabzb is dsaU þ aab sbU : This is in T1,0M if and only if it has no T0,1M component. To see what this N implies, remember that j(s) is the section of (U  Cm) T*M that sends x to (x, dsU þ a sU). What is written above is one of the Cm coordinates of the vector valued 1-form dsU þ a sU. Thus, s is holomorphic if and only if dsU þ a sU has no T0,1M part. This verifies the claim that s defines a holomorphic map from ¯ M to E if and only if @ s ¼ 0. Such a section of E is said to be a holomorphic section. The notion of a holomorphic section is useful for the following reason: Proposition 18.9 Suppose that s is a holomorphic section of E with the property that @ s defines a surjective linear map from T1,0M to E at each point in s1(0). Then s1(0)  M is a holomorphic submanifold. Proof of Proposition 18.9 Fix a point p 2 s1(0) and fix an open neighborhood U  M around p with an isomorphism j: EjU ! U  Cm. Write j(s) as the map x ! (x, sU(x)). Viewed as a map between real manifolds, this map is transversal to the submanifold U  {0} if and only if the map sU: U ! Cm ¼ R2m has surjective differential along s1 U (0). The covariant derivative of s is such that N j(s) is the section x ! (x, dsU þ a sU) of (U  Cm) T*M. Where sU ¼ 0, this is just x ! (x, dsU). Thus, sU is transverse to 0 if and only if dsU is a surjective  map from TM to Cm ¼ R2m along s1 U (0). If @ sU ¼ 0, then a look at the real and imaginary parts of @ sU finds that surjectivity of dsU as a map from TM to R2m occurs if and only if @ sU is surjective as a linear map from T1,0M to Cm. Assuming that this is the case, then it follows that the tangent space to s1(0) appears at a point p 2 U as the vector subspace of T1,0M that is sent to zero by @ sU. This being the case, it follows that the almost complex structure of M preserves this subspace, and so s1(0) is a holomorphic submanifold.

18.7

18.7

Example on CPn

Example on CPn

View CPn as S2nþ1/S1 were l 2 S1 acts on the unit sphere S2nþ1  Cnþ1 so as to send a given point (z1, . . . , znþ1) to (lz1, . . . , lznþ1). Here, S1 is identified with the unit circle in C. As noted in Chapters 10.5 and 10.6, the quotient map p: S2nþ1 ! S2nþ1/S1 ¼ CPn defines a principal S1 bundle. A connection, A, on this principal S1-bundle was described in Example 14.10. By way of reminder, the connection 1-form is the restriction to S2nþ1 of the 1-form 1 A ¼ ∑1anþ1 ðza dza  za dza Þ 2 on Cnþ1. The pull-back of its curvature 2-form, p*FA, is thus the restriction to S2nþ1 of the form p*FA ¼ ∑1anþ1 dza ∧ dza : Keeping this in mind, recall from Example 17.11 that CPn has a Ka¨hler structure whose Ka¨hler form, oFS, pulls back to S2nþ1 via p as the restriction from Cnþ1 of the form i 2 i jzj dza ∧dza  jzj2za dza ∧ zb dzb : 2 2 Here, repeated indices are implicitly summed over their values in the set {1, . . . , nþ1}. As explained next, the restriction of the latter to TS2nþ1 is equal to  2i p*FA . The way to see this is to note that the right-most term in the expression above for p*oFS is zero when evaluated on any two tangent vectors to S2nþ1. Here is why: The 1-form djzj2 is zero on TS2nþ1, and the latter written out is zad z¯a þ z¯a dza. Thus, zad z¯a ¼  z¯a dza on TS2nþ1. As a consequence, the pull-back to TS2nþ1 of the right-most term above is the wedge of a vector with itself. Granted this identification of pull-backs, it follows that FA ¼  2i oFS , and so FA is of type (1, 1). Note in this regard that Chapter 17.7 exhibits oFS explicitly in complex coordinates which make manifest the fact that it is a 1–1 form. By the way, it is not necessary to go through the exercise of writing out the Ka¨hler form to verify that it is of type (1, 1). This is because the Ka¨hler form on any Ka¨hler manifold is automatically of type (1, 1). This can be seen as follows: Take an orthonormal frame for T*M at any given point of the form {e1, . . . , e2n} such that Je2k1 ¼ e2k and Je2k ¼ e2k1. Then P o ¼ 1an e2a1 ∧ e2a. Meanwhile {ua ¼ e2a1 þ ie2a}1an is a basis for P T1,0M, and this understood, it follows that o ¼ 2i 1an ua ∧ u a:

279

280

18 : Holomorphic submanifolds, holomorphic sections and curvature

It follows from the fact that FA is of type (1, 1) that the induced covariant derivative A on any complex bundle associated to the principal bundle p: S2nþ1 ! CPn has curvature 2-form of type (1, 1). By way of an example, fix a nonnegative integer p, and consider the representation rp: S1 ! Gl(1; C) that sends l to lp. The corresponding complex line bundle is then the equivalence class of points ((z1, . . . , znþ1), Z) 2 S2nþ1  C under the equivalence relation that identifies the latter with ((lz1, . . . , lznþ1), lpZ). I use Ep ! CPn to denote this complex line bundle. A section of Ep is a map h: S2nþ1 ! C that obeys hðlz1 ; . . . ; lzn Þ ¼ lp hðz1 ; . . . ; zn Þ: For example, the restriction to S2nþ1 of a homogeneous polynomal of degree p on Cnþ1 defines a section of Ep. As explained next, such polynomials define holomorphic sections of Ep. To prove this assertion, I will take its directional derivative along the horizontal lifts of vectors from T0,1CPn. To see what the latter look like, start with the vector field @@z on Cnþ1. Its projection to TS2nþ1 is a   @ 1 @  @ : a ¼  za zb þZ s b @za 2 @zb @zb The added term is to make s¯ a jzj2 ¼ 0. This vector field on S2nþ1 is not horizontal. Its horizontal projection is     @ @  a zb a  A s :  zb s @zb @zb Now, Að saÞ ¼  12 Za , so what is written above is     @ 1 @ @ 1 @ @ @ @ þ za zb ¼  za zb þ zb  zb  zazb : @za 2 @zb @zb 2 @zb @zb @za @zb As I next demonstrate, the collection f@@z a  zazb @@z b g1anþ1 of (n þ 1) vector fields on S2nþ1 span the horizontal lift of T0,1CPnþ1. To see that this is the case, it is enough to prove that these are annihilated by the pull-back via the projection p of 1-forms from T1,0CPn. Consider, for example, the part of S2nþ1 over the open set O nþ1  CPn which is the image of the subset where znþ1 6¼ 0. This set has complex coordinates (w1, . . . , wn) whose pull-back to Cnþ1{0} is (z1/znþ1, . . . , zn/znþ1). The 1-forms {dw1, . . . , dwn) span T1,0CPn 1 1 over O nþ1, and so their pull-backs are {znþ1 (dza  zaznþ1 dznþ1)}1an. Each of these annihilates all vectors in the set f@@z a  zazb @@z b g1anþ1. Each of the vector fields in this set annihilates the polynomial h because h is the restriction to S2nþ1 of a polynomial in the complex coordinates {za} on Cnþ1. Thus, h is a holomorphic section of Ep.

Additional reading

Additional reading  Complex Geometry: An Introduction, Daniel Huybrechts, Springer, 2004.  Lectures on Ka¨hler Geometry, Andrei Moroianu, Cambridge University Press, 2007.  Algebraic Curves and Riemann Surfaces, Rick Miranda, American Mathematical Society, 1995.  Principles of Algebraic Geometry, Phillip Griffiths and Joseph Harris, WileyInterscience, 1994.  Undergraduate Algebraic Geometry, Daniel Perrin, Cambridge University Press, 1989.  Complex Algebraic Curves, Frances Kirwan, Cambridge University Press, 1992.

281

19

The Hodge star

The holomorphic constraint for a section of a holomorphic bundle on a complex manifold is an example of a differential equation constraint on a section of a vector bundle. To describe perhaps the first such constraint, suppose now that M is a smooth manifold. The exterior derivative, d, is a linear map that takes a section of ∧p T*M to a section of ∧pþ1T*M. The p’th De Rham cohomology is, by definition, the vector space that is obtained by defining an equivalence relation on the kernel of d that posits o o0 when o0 differs from o by a form in the image of d. One can ask whether there exists a reasonably canonical choice of p-form to represent any given equivalence class. As explained momentarily, if M is compact, oriented and has a Riemannian metric, then there exists a canonical representation of each equivalence class.

19.1

Definition of the Hodge star

To set the stage, let n denote the dimension of M and let volM denote the metric’s volume form as defined in Chapter 15.1. The first remark is that the metric with its associated volume form together supply a bundle isomorphism p np *: ∧ T*M ! ∧ T *M

which is defined so that u∧*u ¼ juj2 volM: Here, j · j denotes the norm on ∧p T*M that is defined by the given metric. More generally, if u and u0 are any two p-forms, use hu, u0 i to denote their inner product as defined by the metric. Then u ∧*u0 ¼ hu; u0 i volM : Some linear algebra can be used to see that this does indeed characterize *. To give some idea as to what * looks like, fix an oriented, orthonormal basis

19.2

Representatives of De Rham cohomology

{e1, . . . , en} for T*M at a given point. The volume form at this point is then e1 ∧ . . . ∧ en. This understood, then 1 p pþ1 n *ðe ∧ ∧e Þ ¼ e ∧ ∧e :

Other basis elements of ∧p T*M have the form eil ∧ . . . ∧ eip where 1  i1 < . . . < ip  n. The image of this p-form under * is ð1Þej1 ∧ ∧ejnp where j1 < . . . < jn-p is the ordered sequence of integers obtained from {1, . . . , n} by deleting {i1, . . . , ip}. The sign, þ1, or 1, is determined by the parity of the number of transpositions needed to reorder the ordered set {i1, . . . , ip, j1, . . . , jn-p} as {1, . . . , n}. The definition that two successive applications of the Hodge star defines an involution of ∧p T*M, in particular *2 ¼ (1)p(n-p). For example, * sends the volume form to the constant function 1 (this is a zero form on M). As a second example, note that in the case dim(M) ¼ 2, the Hodge * maps T*M to itself with square *2 ¼ 1 on 1-forms. This being the case, it defines an almost complex structure on the tangent space when the metric is used to identify TM with T*M. This is the same almost complex structure that was introduced using the metric in Chapter 17.4.

19.2

Representatives of De Rham cohomology

Continue with the assumption that M is a compact, oriented manifold with a Riemannian metric. The Hodge theorem can be used to find a canonical representative of each De Rham cohomology class. What follows is the result: Theorem 19.1 Let M denote a compact, oriented, Riemannian manifold. Fix p 2 {0, . . . , n} and let z 2 HpDe Rham(M) denote a given class. Then there exists a unique p-form o that obeys both do ¼ 0 and d*o ¼ 0, and whose De Rham cohomology class is z. A form o that obeys both do ¼ 0 and d*o ¼ 0 is said to be harmonic. Let Hp denote the vector space of harmonic p-forms. This theorem asserts that HpDe Rham(M) is isomorphic to Hp. Because * maps Hp to Hnp, and because o∧ * o ¼ joj2 volM, this theorem has the following corollary: Corollary 19.2 Let M denote a compact, oriented n-dimensional manifold. Then p np each p 2 {0, 1, . . . , n} version of HDe Rham(M) is isomorphic to HDe Rham(M). In particular, there is an isomorphism with the following property:

283

19 : The Hodge star

Ð p np if z 2 HDe Rham(M) and *z is its partner in HDe Rham(M), then M oz ∧o*z > 0 for any pair oz and o z of representative forms. * The corollary asserts the De Rham cohomology version of Poincare´ duality.

19.3

A fairy tale

Before stating the Hodge theorem, consider for the moment the following fairytale scenario: Let V and V0 denote finite dimensional vector spaces with inner product, and let D: V ! V0 denote a linear map. Since both V and V0 have inner products, there is a well-defined adjoint map D{: V0 ! V. It is defined by the following rule: hv; D{ v0 iv ¼ hDv; v0 iv0 for all pairs v 2 V and v0 2 V0. Here, h , iV and h , iV0 denote the respective inner products on V and V0. What follows is a fundamental observation: 0

0

Let p: V ! V=DðVÞ denote the quotient projection: The restriction of 0

p to the kernel of D{ maps this kernel isomorphically onto V=DðVÞ: To see that p is 1–1 on the kernel of D{, suppose that v0 2 V0 is such that both D{v0 ¼ 0 and p(v0 ) ¼ 0. If p(v0 ) ¼ 0, then v0 ¼ Dv for some v 2 V. As a consequence D{Dv ¼ 0 and so hv, D{DviV ¼ 0. But according to the definition of D{, this says the fact that hDv, DviV0 ¼ 0, and so v0 ¼ 0. To see that p maps the kernel of D0 surjectively onto V0 /D(V), suppose that z is an element in the latter space, and let z0 2 V0 denote an element that projects to z. Let v0 denote the orthogonal projection of z0 onto the D(V). Then p(v0  z0 ) ¼ z. Meanwhile, hDv, v0 iV0 ¼ 0 for all v 2 V, which says D{v0 ¼ 0. To continue the fairy tale, suppose that V ¼ V0 and that D2 ¼ 0. Note that this implies that (D{)2 ¼ 0 also. Let H denote the vector subspace {v 2 V : Dv ¼ 0 and D{v ¼ 0}. Thus, H ¼ ker(D) ker(D{). With the preceding as background, what follows is the finite dimensional analog of the Hodge theorem: ¨

284

There is an orthogonal decomposition V ¼ imageðDÞ

M

imageðD{ Þ

M

H:

Indeed, it follows from the definition of D{ that the three summands are pairwise orthogonal. Two applications of the argument given two paragraphs back proves that they account for the whole of V. Indeed, a direct application L asserts that there is the orthogonal decomposition V ¼ im(D) ker(D{). { 2 { Since (D ) ¼ 0, the image of D is a subspace of the kernel of D{. This

19.4

The Hodge theorem

understood, apply the argument of the preceding paragraph with V now replaced by the kernel of D{ and D{ then replaced by D’s restriction to the kernel of D{. The argument in this context gives the orthogonal decomposition L ker(D{) H.

19.4

The Hodge theorem

To see the relevance of this fairy tale, imagine replacing V with the vector space L of *(M) ¼ 0pn C1(M; ∧p T*M). Thus, the vector space of differential forms of all degrees. Define the inner product on this space as follows: If a and b have different degrees, then their inner product is zero. If they have the same degree, then their inner product is ð a ∧*b: M

This inner product on any summand C1(M; ∧p T*M) in * is called the L2-inner product. To complete the fairy-tale analogy, replace the fairy-tale’s linear map D with the exterior derivative d. With regards to this analogy, remark that the adjoint of the exterior derivative d as defined by this inner product is denoted by d{. It is usually called the formal L2 adjoint of d. An application of Stoke’s theorem finds that d{ ¼ ð1Þnp *d* on the vector space of p-forms. Note that d{ maps p-forms to (p  1)-forms and is such that (d{)2 ¼ 0. Theorem 19.3 (The Hodge theorem) Let M denote a compact, oriented Riemannian manifold. Then there is an orthogonal (with respect to the L2-inner product) decomposition M M imageðd{ Þ H*: * ¼ imageðdÞ

This means the following: Let p 2 {0, 1, . . . , n} and let o denote any given p-form. Then there is a unique triple (a, b, g) of (p1)-form a 2 im(d{), of (p þ 1)-form b 2 im(d), and harmonic p-form g such that o ¼ da þ d{b þ g. The proof that there is at most one such (a, b, g) is obtained by copying the argument from the fairy tale. The proof of existence requires various tools from analysis that are beyond the scope of this book. Even so, I can pinpoint the reason for the complications: The space * with its L2 metric is not a complete inner product space.

285

286

19 : The Hodge star

19.5

Self-duality

Suppose that M is an oriented, Riemannian manifold with even dimension 2n. Then * acts on ∧n T*M as an involution with square (1)n. If n is even, this is 1, L n and so ∧n T*M decomposes as the orthogonal direct sum nþ  , where nþ consists of the forms o with *o ¼ o, and n consists of the forms o with *o ¼ o. Each of these summands is a real vector bundle over M with fiber ð2nÞ! nþ are said to be self-dual and those in nþ are said dimension 2ðn!Þ 2 . Forms in  to be anti-self-dual. Because * maps the space Hn of harmonic n-forms to itself, this vector space L also has a pointwise orthogonal direct sum splitting as Hn ¼ Hn Hnþ where Hnþ consists of forms o with do ¼ 0 and *o ¼ o. Meanwhile, Hn consists of the closed, anti-self-dual n-forms. Neither the dimension of Hnþ nor that of Hn depends on the choice of metric. They depend only on the smooth strucp ture of M. Here is why: The De Rham cohomology HDe Rham(M) has the 0 symmetric pairing that associates to classes z and z the number ð 0 0 Qðz; z Þ ¼ o ∧ o M

as computed using any pair of representative closed n-forms o and o0 for z and z0 . As seen earlier in the chapter, the pairing is nondegenerate. Let nþ denote the dimension of the maximal vector subspace in HnDe Rham(M) on which Q is positive definite. This is the dimension of Hnþ. The dimension of Hn– is that of the maximal subspace on which Q is negative definite. (As it turns out, the dimensions of Hnþ and Hn– depend only on the underlying topological structure of M.) The difference nþ  n is called the signature of M. Of special note with regards to self-duality is the case n ¼ 2 and so dim(M) ¼ 4. In this case, the self-duality condition has been used extensively to find canonical covariant derivatives on bundles over M. To say more, suppose that p: E ! M is a vector bundle and  is a covariant derivative on N M. The curvature 2-form of  is a section, F, of End(E) ∧2 T*M. As such, it N 2þ can be written as (F)þ þN(F)– where (F)þ is the part in End(E)  and (F)– is the part in End(E) 2–. The covariant derivative is said to be anti-selfdual when (F)– ¼ 0; and it is said to be anti-self-dual when (F)þ ¼ 0. As it turns out, these are very stringent conditions on a covariant derivative. Simon Donaldson discovered how the spaces of self-dual connections can be used in certain cases to distinguish distinct smooth structures on a given topological 4-dimensional manifold. For more about this fantastic discovery, I recommend his book with Peter Kronheimer, The Geometry of Four-Manifolds (1997).

Additional reading

Additional reading  Riemannian Geometry and Geometric Analysis, Ju¨rgen Jost, Springer, 2008.  The Geometry of Differential Forms, Shigeyuki Morita, American Mathematical Society, 2001.  The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds, Steven Rosenberg, Cambridge University Press, 1997.  Instantons and Four Manifolds, Daniel Freed and Karen Uhlenbeck, Springer, 1990.  The Geometry of Four-Manifolds, Simon Donaldson and Peter Kronheimer, Oxford University Press, 1997.

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List of lemmas, propositions, corollaries and theorems

Theorem 1.1: The inverse function theorem Theorem 1.2: The implicit function theorem Theorem 1.3: Sard’s theorem Lemma 1.5: Submanifolds and injective differentials Theorem 1.6: The contraction mapping theorem Lemma 2.1: Subgroups of Lie groups are Lie groups Lemma 3.7: Characterization of linear maps Proposition 5.4: Vector bundles and pull backs from Grassmannians Proposition 5.7: Transversal maps and submanifolds Lemma 6.4: Almost complex structures and complex vector bundles Lemma 7.2: Orientable bundles and the determinant line bundle Lemma 7.3: SU(n) bundles and the determinant line bundle Theorem 8.1: The geodesic theorem Theorem 8.2: The vector field theorem Proposition 8.3: The existence of geodesics Proposition 8.4: Families of geodesics Proposition 8.5: Characterization of geodesics on SO(n) Proposition 8.6: Characterization of geodesics on Gl(n; R) Proposition 8.7: Characterization of geodesics on Gl(n; C) Proposition 8.8: Characterization of geodesics on U(n) and SU(n) Proposition 8.9: Characterization of geodesics on compact matrix groups Proposition 9.1: The existence of the exponential map Proposition 9.2: Characterization of Gaussian coordinates Lemma 9.3: Uniqueness of short geodesics Lemma 9.4: Length minimizing curves and geodesics Proposition 10.1: Manifolds given by quotients of compact Lie groups by subgroups Lemma 11.1: Characterizing bundle homomorphism Theorem 11.2: Pull-backs of principal bundles by homotopic maps Corollary 11.3: Principal bundles over contractible spaces Corollary 11.4: Pull-backs of bundles over surfaces via maps to S2

290

List of lemmas, propositions, corollaries and theorems

Proposition 12.1: Closed forms on contractible spaces Corollary 12.2: Poincare´ lemma for closed forms Theorem 13.1: Frobenius theorem for foliations Theorem 13.2: Classification theorem for flat connections Theorem 14.3: The ad-invariant function theorem Proposition 14.7: Integration and the degree of a map Lemma 14.8: Variation of almost complex structures Proposition 14.12: Integration of n-forms and De Rham cohomology Corollary 14.13: Top dimensional De Rham cohomology Theorem 14.14: The n-to-n Sard’s theorem Lemma 14.15: Integral of pull-back of volume form Theorem 15.1: The Levi-Civita theorem Proposition 16.4: The Gauss–Bonnet formula for surfaces Theorem 16.5: Classification of metrics on surfaces Theorem 16.6: Universal covers and nonpositive sectional curvature Proposition 16.7: Characterization of Jacobi fields Proposition 16.8: The exponential map and Jacobi fields Proposition 16.9: Constant sectional curvature metrics Proposition 16.11: Curvature on compact matrix groups Lemma 16.12: Shur’s lemma Theorem 17.5: Newlander–Nirenberg theorem Theorem 17.6: Complex structures on Riemann surfaces Proposition 17.7: Covariantly constant almost complex structures Proposition 17.8: Closed almost Ka¨hler forms and complex structures Proposition 18.1: Holomorphic submanifolds and complex submanifolds Proposition 18.2: Holomorphic submanifolds of CPn Lemma 18.3: The zero locus of a holomorphic function Lemma 18.4: Holomorphic submanifolds of Cn Proposition 18.7: The curvature of a Ka¨hler metrics Proposition 18.8: Curvature of type (1.1) and holomorphic bundles Proposition 18.9: The zero locus of a holomorphic section Theorem 19.1: De Rham cohomology and harmonic forms Corollary 19.2: Poincare´ duality for De Rham cohomology Theorem 19.3: The Hodge theorem

List of symbols

∧*: 68 ∧: 46, 68 ∧k E* when E is a vector bundle: 46 ∧k V* when V is a vector space: 46 ∧k: 46, 68 N E* when E is a vector bundle: 44 Nk k V* when V is a vector space: 44 ^*A: 137 f Nj: 252 @ @m: 21 @ @mij : 21 ℓª: 78 *: 282, 283 @ƒ: 247 a ∧ a for matrix valued 1-forms: 145 a ∧ b for End(E) valued forms: 145 A(n; ℂ): 35 A(n; R): 35 A0(n; ℂ): 56 Aut(P): 156 c(tm): 175 ch(tm): 175 ck(E): 178 ℂ*: 248 ℂℙn: 67 d▽: 144 d: 139, 140 det(·): 14 det(m): 14 detℂ: 21 detR: 20

292

List of symbols

det(E) for real bundles: 46 det(E) for complex bundles: 69 dist(·, ·): 78 dm 17 dmij: 16 dmT: 21 E¯: 62 L 0 E E when E and E0 are vector bundles: 44 N 0 E E when E and E0 are vector bundles: 43 E* when E is a vector bundle: 41 E/E0 : 40 Eℂ: 62 em(·): 53, 56 ER: 60 FA: 149 (FA)U: 149 F▽: 145, 148 F ▽: 273 ƒ(*(FA)): 176 ƒk(*(FA)): 176 FM;G: 159 G/H: 108 Gj: 18 Gl(n; R): 14 Gl(n; ℂ): 17 Gr(m; n): 10 Grℂ(m; n): 65 hA,ª(p): 160 HA: 131 Herm(n): 20 Hom(E; E0 ) when E and E0 are vector bundles: 42 0 Hom(Rn; Rn ): 34 Hom(V; V0 ) when V and V0 are a vector space: 42 Hom(1(M); G)/G: 158 Hom(1(M); G): 158 HpDe Rham(M): 141 j0: 18, 245 j for complex bundles: 59 j and j0: 61 ker(*): 130 lie(G): 92 Lv: 144

List of symbols

M(n; R): 14 M(n; ℂ): 17 Mj: 18 mT: 17 Mv: 110 O(n): 17 oˆ, the zero section: 26 P  V: 119 pk(E): 180 R: 218 R+: 248 Rabcd: 217 Ricab: 218 Riem: 218 Rℙn: 9 Sl(n; R): 16 Sl(n: ℂ): 19 SO(n): 17 sP(, FA): 172 SU(n): 20 Sym(n; R): 17 Sym+(m; R): 264 Symk(E*) when E is a vector bundle: 45 Symk(V*) when V is a vector space: 45 T1,0 for a surface: 64 T1,0M and T0,1M: 245 T1,0M and T0,1M: 245 T2n: 249 Tp,qM: 245 trace(·): 14 tr(·): 14 tr(·): 172 T¸: 221 U(n): N 0 V V when V and V0 are vector spaces: 43 L 0 V V when V and V0 are vector spaces: 43 V* when V is a vector space: 41 jkm: 80 M: 208 : 128 : E ! M. 1(M): 158

293

294

List of symbols

*: 1–9 f*▽: 138 c*E: 48 M: 144

Index

Abelian group: 15 action of C1(M): 125 ad: 132 ad-invariant function theorem: 176 ad-invariant: 175 adjoint representation: 121, 132 almost all values: 271 almost complex structure: 18 almost complex structure for a bundle: 59 almost complex structure on S6: 255 almost Ka¨hler form: 256 anti-self dual covariant derivative: 286 anti-self dual: 286 antisymmetric powers: 45 associated fiber bundle: 156 associated vector bundle: 119 Aut(P): 156 basis of sections: 36 Bianchi identity: 170 bi-invariant metric: 86 Brieskorn sphere: 115 bundle projection map: 26 bundle transition function: 27 Cauchy–Riemann equations: 246 Characteristic class for a complex vector bundle: 178 Characteristic class for a principal bundle: 175 Characteristic form: 171, 172 Chern character classes: 179 Chern class: 178 Chern–Simons form: 190 Christoffel symbols: 80 classification theorem for metrics on 2-dimensional manifolds: 229 closed form: 141 cocycle constraint: 27 cocycle data: 28

cocycle definition for a complex bundle: 60 cocycle constraints for a principal bundle: 105 commutator of vector fields: 148 commuting group elements: 158 compact matrix group: 92 compatible almost complex structure: 261 complex conjugate bundle: 63 complex frame bundle: 107 complex Grassmannian: 65 complex manifold structure: 247 complex projective space: 67 complex structures on S1  S3: 259, 260 complex structures on S2p1  S2q1: 260 complex vector bundle: 59 complexification: 62 conformal: 229, 230 conjugate: 158 connection on a principal bundle: 131 constant scalar curvature: 227 contractible set: 143 contraction mapping theorem: 12 coordinate atlas: 1 coordinate transition function: 2 cotangent bundle: 33 covariant derivative: 125 covariantly constant: 206, 212 covering space: 159 curvature 2-form: 149 curvature of a connection: 149 curvature of a covariant derivative: 145 curvature of a Ka¨hler metric: 272ff curvature of type (1,1): 275 curvature operator: 217 De Rham cohomology, homotopy invariance: 142 De Rham cohomology: 141 decomposable element: 43

296

Index

degree of a map: 182 degree of a map, homotopy invariance: 182 derivation: 36 diffeomorphic manifolds: 3 diffeomorphism: 3 differential of a map: 3 direct sum: 43 directional derivative: 37 dist(·,·): 78 dual bundle: 41 dual vector space: 41 Einstein equations: 227 Einstein tensor: 218 endomorphism between vector bundles: 34 equivalence class: 8 equivalence relation: 8 Euclidean metric on Rn: 82 exact form: 141 exact sequence: 130 exotic sphere: 115 exponential map: 97 exponential map on M(n; C): 56 exponential map on M(n; R): 53 exterior covariant derivative: 144 exterior derivative: 139, 140, 141 exterior product: 46 exterior product construction: 68 exterior product construction of complex bundles: 68

Gaussian coordinates: 98 geodesic: 80 geodesic equation: 80 geodesic theorem: 79 geodesically complete: 96 geodesics on matrix groups: 92 Geometrization conjecture: 239 Geometrization theorem: 239 Grassmannian: 10 Grassmannian of n-planes in Rm: 10 Grassmannian of n-planes in Cm: 65 harmonic form: 283 Heisenberg group: 239 Hermitian metric: 72 Hermitian metric on T1,0M: 255 Hodge star: 282 Hodge’s theorem: 283 holomorphic differential: 269 holomorphic function: 246 holomorphic map: 277 holomorphic section: 277 holomorphic submanifold: 268 holomorphic vector bundle: 275 holomorphically equivalent: 275 holonomy: 160 homogeneous polynomial: 269 homomorphism of a vector bundle: 34 horizontal foliation: 155 horizontal lift: 135 horizontal subbundle: 131 horizontal vector: 134 hyperbolic metric on Rn: 84 hyperbolic metric: 222

fiber: 26 fiber bundle: 114 fiber dimension: 25 flat connection: 152 flat connection, classification theorem: 159 flat metric: 222 foliation: 155 form of type (p, q): 245 formal, L2 adjoint: 285 frame bundle: 106 free group: 158 Frobenius theorem: 155 Fubini–Study metric: 261 fundamental group: 157, 158

identity element, Ø: 15 immersion: 7, 57 implicit function theorem: 4 induced metric (for codimension 1 submanifolds): 83 injectivity radius: 98 integration on a manifold: 192ff inverse: 15 inverse function theorem: 3 involutive: 155 irreducible representation: 243 isomorphic principal bundles: 104 isomorphism between vector bundles: 34

gauge equivalent connections: 157 gauge group: 156 Gauss map: 48 Gauss–Bonnet formula: 227

Jacobi field equation: 233, 234, 235 Jacobi field: 234 Jacobi relation: 109 JSJ decomposition: 239

Index

k’th Pontryagin class: 180 Ka¨hler manifold: 256, 257 Ka¨hler metric, restriction to holomorphic submanifold: 269 k-form: 46 k’th Chern class: 178 L2-inner product: 285 lattice: 221 leaf of a foliation: 155 left invariant: 52 left invariant forms, vector fields: 52, 55 left translation: 52 Leibnitz’s rule: 125 length minimizing: 79 length of a path: 78 lens space: 9 Levi-Civita connection: 210 Levi-Civita theorem: 210 Lie algebra: 108 Lie bracket: 109 Lie derivative of a differential form: 144 Lie derivative: 37 Lie group: 15 linear map, characterization: 27 local trivialization: 26 locally conformally flat: 232 locally finite coordinate atlas: 1 locally finite: 1 locally length minimizing: 80, 100 Lorentz group: 84, 222 Lorentz metric: 222 maximal extension: 96 metric: 72 metric compatible covariant derivative: 205 metric topology: 79 Mobius bundle: 28 Mobius bundle and orientability: 74 multiplication map: 14 Newlander–Nirenberg theorem: 252 Nijenhuis tensor: 252 normal bundle: 41 normal coordinates: 98 octonion algebra: 254 1-form: 38 orientable: 73 orientable bundles and det(·): 73 orthonormal frame bundle: 107

parallel along a curve: 213 parallel transported along a curve: 213 partition of unity: 13 path: 78 Pauli matrices: 63 Poincare´ duality: 284 Poincare´ lemma: 143 Pontryagin class: 180 positive/negative curvature operator: 226 positive/negative sectional curvature: 226 principal bundle: 104 principal bundles and homotopic maps: 136 product bundle: 28 projection map: 26 pseudoholomorphic: 277 pull-back homomorphism: 50 pull-back of a connection: 137 pull-back of a covariant derivative: 138 pull-back of a principal bundle: 116 pull-back of bundles: 48 pull-back of complex bundles: 70 pull-back of differential forms: 50 pull-back of principal bundles from Grassmannians: 121 pull-back of sections: 48 pull-backs of vector bundles from Grassmannians: 50 pull-backs of vector bundles from complex Grassmannians: 70 push-forward of vector fields: 51 quaternion algebra: 63 quaternionic projective space: 116 quotient bundle: 40 quotients of Lie groups: 108 quotients of manifolds: 8 real projective space: 9 real vector bundle: 25 reducible principal bundle: 118 reducible representation: 243 refinement of an open cover: 34 regular value: 3 representation a Lie group: 109 Ricci tensor: 218 Riemann curvature tensor: 217 Riemannian manifold: 77 Riemannian metric: 77 right invariant forms, vector fields: 52, 53, 55 right invariant: 52

297

298

Index

right translation: 52 round metric on Sn: 82 Sard’s theorem, proof for maps between n-dimensional manifolds: 198 Sard’s theorem: 4, 198 scalar curvature: 218 Schwartzchild metric: 223 section of a vector bundle: 35 sectional curvature: 232 sectional curvature theorem: 232 self dual: 286 self dual covariant derivative: 286 Shur’s lemma: 243 signature of a manifold: 286 simple Lie group: 243 simply connected manifold: 158 smooth manifold: 2 smooth map: 2 smooth structures on S7: 114 smoothing a map: 164, 165 solve geometry: 239 speed of a path: 78 spherical coordinates, in the Gaussian context: 100 spherical coordinates: 100 stabilizer: 109 standard metric on Rn: 82 subbundle: 39 subgroup: 16 submanifold: 4, 7 submersion: 8, 57 super algebra: 144 symmetric powers: 45 symplectic form: 256 tangent bundle: 29, 30 tautological bundle over Gr(m; n): 29 tautological bundle n-plane bundle over GrC(m; n): 65 tautological bundle over Rn: 28, 75

tensor: 121 tensor bundle: 121 tensor field: 121 tensor powers: 44 tensor product bundles: 43 the canonical almost complex structure on Cn: 245 topological manifold: 1 tori: 221 torsion: 209 torsion free covariant derivative: 209 torus: 221, 229 totally geodesic submanifold: 87 trace: 14 transition function for a principal bundle: 105 transversal: 57 transversality and submanifolds: 57 trivial bundle: 28 trivial principal bundle: 104 universal characteristic class: 178 universal classifying space: 136 universal covering space: 159 vanishing Riemann curvature: 226 vector bundle endomorphism, characterization of: 126 vector field theorem: 81 vector field: 36 vertical subbundle: 131 volume from 208 volume of an open set: 208 volume of M: 208 volume, definition of: 198 wedge product: 46 zero section, oˆ: 26 ø-compatible: 256